The psychologist Robert M. Gagne has
done the research into the phases of learning sequence and the types of
learning. His research is particularly relevant for teaching mathematics.
Professor Gagne has used mathematics as a medium for testing and applying his
theories about learning and has collaborated with the University of Maryland
Mathematics Project in studies of mathematics learning and curriculum
development.
A.
The
Objects of Mathematics Learning
There are two objects
of mathematics learning. They are direct and indirect things which we want
students to learn in mathematics. The direct objects of mathematics learning
are facts, skills, concepts, and principles. Some of the many indirect objects
are transfer of learning, inquiry ability, problem-solving ability,
self-discipline, and appreciation for the structure of mathematics. The direct
objects of mathematics learning - facts, skills, concepts, and principles - are
the four categories which mathematical content can be separated.
Below are the descriptions of each direct
object of mathematics learning.
1. Mathematical
Facts
Mathematical
facts are those arbitrary conventions in mathematics such as the symbols of
mathematics. It is a fact that 2 is the symbol for the word two, that + is the
symbol for the operation of addition, and that sine is the name given to a
special function in trigonometry. Facts are learned through various techniques
of rote learning such as memorization, drill, practice, timed tests, games, and
contests. People are considered to have learned a fact when they can state the
fact and make appropriate use of it in a number of different situations.
2. Mathematical
Skills
Mathematical
skills are those operations and procedures which students and mathematicians
are expected to carry out with speed and accuracy. Many skills can be specified
by sets of rules and instructions or by ordered sequences of specific
procedures called algorithms. Among the mathematical skills which most people
are expected to master in school are long division, addition of fractions and
multiplication of decimal fractions. Constructing right angles, bisecting
angles, and finding unions or intersections of sets of objects and events are
examples of other useful mathematical skills. Skills are learned through
demonstrations and various types of drill and practice such as worksheets, work
at the chalkboard, group activities and games. Students have mastered a skill
when they can correctly demonstrate the skill by solving different types of
problems requiring the skill or by applying the skill in various situations.
3. Mathematical
Concepts
A concept in mathematics
is an abstract idea which enables people to classify objects or events and to
specify whether the objects and events are examples or non-examples of the
abstract idea. In this, the examples of concepts are sets, subsets, equality,
inequality, triangle, cube, radius, and exponent. A person who has learned the
concept of triangle is able to classify sets of figures into subsets of
triangles and non-triangles. Concepts can be learned either through definitions
or by direct observation.
4. Mathematical
Principles
The most complex
of the mathematical objects are principles. Principles are sequences of
concepts together with relationships among these concepts.
The following statements are the
examples of principles.
Ø The
square of the hypotenuse of a right triangle is equal to the sum of the squares
of the other two sides.
Ø Two
triangles are congruent if two sides and the included angle of one triangle are
equal to two sides and the included angle of the other.
Principles can
be learned through processes of scientific inquiry, guided discovery lessons,
group discussions, the use of problem solving strategies, and demonstrations. A
student has learned principles when he or she can identify the concepts included
in the principle, put the concepts in their correct relation to one another,
and apply the principle to a particular situation.
As a mathematics
teacher, we should develop testing and observation techniques to assist us in
recognizing students’ viewpoints of the concepts and principles which we are
teaching. All of us have at times memorized the proofs of theorems, with no
understanding of the concepts and principles involved in the proof, in order to
pass tests. While this subterfuge is a form of learning, it is not what teacher
hope to have students learning by proving theorems.
B.
The
Phases of A Learning Sequence
There are four
phases of a learning sequence. They are the apprehending phase, the acquisition
phase, the storage phase, and the retrieval phase.
Below are the descriptions of each phase
of a learning sequence.
1. The
Apprehending Phase
The first phase
of learning is the apprehending phase. It is the learner’s awareness of a
stimulus or a set of stimuli which are present in the learning situation. Awareness,
or attending, will lead the learner to perceive characteristics of the set of
stimuli. What the learner perceives will be uniquely coded by each individual
and will be registered in his or her mind. This idiosyncratic way in which each
learner apprehends a given stimulus results in a common problem in teaching and
learning.
2. The
Acquisition Phase
The next phase
in learning is the acquisition phase. It is attaining or possessing the fact,
skill, concept, or principle which is to be learned. Acquisition of
mathematical knowledge can be determined by observing or measuring the fact
that a person does not possess the required knowledge or behavior before an
appropriate stimulus is presented, and that he or she has attained the required
knowledge or behavior immediately after presentation of the stimulus.
3. The
Storage Phase
After a person
has acquired a new capability, it must be retained or remembered. This is the
storage phase of learning. The human storage facility is the memory, and
research indicates that there are two types of memory. They are short-term
memory and long-term memory.
4. The
Retrieval Phase
The fourth phase
of learning is the retrieval phase. It is the ability to call out the
information that has been acquired and stored in memory. The process of
information retrieval is very imprecise, disorganized, and even mystical.
C.
Types
of Learning
There are eight
types of learning. They are signal learning, stimulus-response learning,
chaining, verbal association, discrimination learning, concept learning, rule
learning, and problem solving.
Below are the descriptions of each type
of learning.
1. Signal
Learning
Signal learning
is involuntary learning resulting from either a single instance or a number of
repetitions of a stimulus which will evoke an emotional response in an
individual.
In order for
signal learning to occur, there must be a neutral signal stimulus and a second,
unexpected stimulus that will evoke an emotional response in the learner which
he or she will associate with the neutral stimulus. In the example of the
person who learned to fear group signing in a first grade music class, the
neutral signal stimulus was singing in a group and the unexpected stimuli were a
shout and a slap.
As a mathematics
teacher, we should attempt to generate unconditioned stimuli which will evoke
pleasant emotions in our students and hope that they will associate some of
these pleasant sensations with the natural signal which is our mathematics
classroom.
2. Stimulus-Response
Learning
Stimulus-response
learning is also learning to respond to a signal. It is voluntary and physical.
Stimulus-response learning involves voluntary movements of the learner’s
skeletal muscles in response to stimuli so that the learner can carry out an
action when he or she wants to do.
Most examples of
pure stimulus-response learning in people are found in young children. They are
learning to say words, carry out various life-supporting functions, use simple
tools, and display socially acceptable behaviors.
3. Chaining
Chaining is the
sequential connection of two or more previously learned non-verbal
stimulus-response actions. The examples of chaining are tying a shoe, opening a
door, starting an automobile, throwing a ball, sharpening a pencil, and
painting a ceiling.
In order for
chaining to occur, the learner must have previously learned each
stimulus-response link required in the chain. If each link has been learned,
chaining can be facilitated by helping the learner establish the correct
sequence of stimulus-response acts for the chain.
Most activities
in mathematics which entail manipulation of physical devices such as rulers,
compasses, and geometric models require chaining. Learning to bisect an angle
with a straightedge and a compass requires proper sequencing and implementing
of a set of previously learned stimulus-response type skills. Among these
skills are the ability to use a compass to strike an arc and the ability to
construct a straight line between two points.
4. Verbal
Association
Verbal
association is chaining of verbal stimuli; that is, the sequential connection
of two or more previously learned verbal stimulus-response actions.
The mental
processes involved in verbal association are very complex and not completely
understood at present. Most researchers do agree that efficient verbal
association requires the use of intervening mental links which act as codes and
which can be either verbal, auditory, or visual images. These codes usually
occur in the learner’s mind and will vary from learner to learner according to
each person’s unique mental storehouse of codes. For example, one person may
use the verbal mental code “y is
determined by x” as a cue for the
word function, another person may
code function symbolically as “y = f(x)”, and someone else may visualize
two sets of elements enclosed in circles with arrows extending from the
elements of one set to the elements of the other set.
The most
important use of the verbal association type of learning is in verbal dialogue.
Good oratory and writing depend upon a vast store of memorized verbal
associations in the mind of the orator or writer. To express ideas and rational
arguments in mathematics it is necessary to have a large store of verbal
association about mathematics.
5. Discrimination
Learning
Discrimination
learning is learning to differentiate among chains; that is, to recognize
various physical and conceptual objects. There are two kinds of discrimination.
They are single discrimination and multiple-discrimination.
As students are
learning various discriminations among chains, they may also be forming these
stimulus-response chains at the same time. This somewhat disorganized learning
situation can, and usually does, result in several phenomena of multiple
discrimination learning (generalization, extinction, and interference).
Ø Generalization
is the tendency for the learner to classify a set of similar but distinct
chains into a single category and fail to discriminate or differentiate among
the chains.
Ø If
appropriate reinforcement is absent from the learning of a chain of stimuli and
responses, extinction or elimination
of that chain occurs.
Ø Interference
can be a problem in learning a foreign language such as French, which has many
words similar in meaning and spelling to English words.
6. Concept
Learning
Concept learning
is learning to recognize common properties of concrete objects or events and
responding to these objects or events as a class.
In order for
students to learn a concept, simpler types of prerequisite learning must have
occurred. Acquisition of any specific concept must be accompanied by
prerequisite stimulus-response chains, appropriate verbal associations, and
multiple-discrimination of distinguishing characteristics. For example, the
first step in acquiring the concept of circle
might be learning to say the word circle as a self-generated stimulus-response
connection, so that students can repeat the word. Then students may learn to
identify several different objects as circles by acquiring individual verbal
association. Next, students may learn to discriminate between circles and other
objects such as triangles and squares. It is also important for students to be
exposed to circles in a wide variety of representative situations so that they
learn to recognize circles which are imbedded in more complex objects. When the
students are able to spontaneously identify circles in unfamiliar contexts,
they have acquired the concept of circle.
7. Rule
Learning
Rule learning is
the ability to respond to an entire set of situations (stimuli) with a whole
set of actions (responses). Rule learning appears to be the predominant type of
learning to facilitate efficient and coherent human functioning. Our speech,
writing, routine daily activities, and many of our behaviors are governed by
rules which we have learned.
In order for
people to communicate and interact, and for society to function in any form
except anarchy, a huge and complex set of rules must be learned and observed by
a large majority of people. Much of mathematics learning is rule learning. For
example, we know that 
and that 
; however without knowing the rule that can be
represented by 
, we would not be able to generalize beyond those few
specific multiplication problem which we have already attempted. In first, most
people learn and use the rule that multiplication is commutative without being
able to state it. In order to discuss this rule, it must be given either a
verbal or a symbolic formulation such as “the order in which multiplication is
done doesn’t make any difference in the answer” or “for all numbers a and b, ”; This particular rule and rules in general, can be
thought of as sets of relations among sets of concepts.
and that ; however without knowing the rule that can be
represented by , we would not be able to generalize beyond those few
specific multiplication problem which we have already attempted. In first, most
people learn and use the rule that multiplication is commutative without being
able to state it. In order to discuss this rule, it must be given either a
verbal or a symbolic formulation such as “the order in which multiplication is
done doesn’t make any difference in the answer” or “for all numbers a and b, ”; This particular rule and rules in general, can be
thought of as sets of relations among sets of concepts.
Mathematics
teachers need to be aware that being able to state a definition or write a rule
on a sheet of paper is little indication of whether a student has learned the
rule. If students are to learn a rule they must have previously learned the
chains of concepts that constitute the rule. The conditions of rule learning
begin by specifying the behavior expected of the learner in order to verify
that the rule has been learned. A rule has been learned when the learner can
appropriately and correctly apply the rule in a number of different situations.
In his book The Conditions of Learning,
Robert Gagne (1970) gives a five step instructional sequence for teaching
rules:
Ø Step 1:
Inform the learner about the form of the performance to be expected when
learning is completed.
Ø Step 2:
Question the learner in a way that requires the reinstatement (recall) of the
previously learned concepts that make up the rule.
Ø Step 3:
Use verbal statements (cues) that will lead the learner to put the rule
together, as a chain of concepts, in the proper order.
Ø Step 4:
By means of a question, ask the learner to “demonstrate” one of (sic) more
concrete instances of the rule.
Ø Step 5:
(Optional, but useful for later instruction): By a suitable question, require
the learner to make a verbal statement of the rule.
8. Problem-Solving
As one might
expect, problem-solving is a higher order and more complex type learning than
rule-learning, and rule acquisition is prerequisite to problem-solving. Problem
solving involves selecting and chaining sets of rules in a manner unique to the
learner which results in the establishment of a higher order set of rules which
was previously unknown to the learner.
Real-word problem solving usually
involves five steps, they are:
Ø Presentation
of the problem in a general form
Ø Restatement
of the problem into an operational definition
Ø Formulation
of alternative hypothesis and procedures which may be appropriate means of
attacking the problem
Ø Testing
hypothesis and carrying out procedures to obtain a solution or a set of
alternative solutions
Ø Deciding
which possible solution is most appropriate or verifying that a single solution
is correct.
D.
Learning
Hierarchies
A learning
hierarchy for problem-solving or rule-learning is a structure containing a
sequence of subordinate and prerequisite abilities which a student must master
before he or she can learn the higher order task. Gagne describes learning as
observable changes in people’s behavior, and his learning hierarchies are
composed of abilities which can be observed or measured.
According to
Gagne, if a person has learned, then that person can carry out some activity
that he or she could not do previously. Since most activities in mathematics
require definable and observable prerequisite learning, mathematics topics lend
themselves to hierarchical analyses. When specifying a learning hierarchy for a
mathematical skill, it is usually not necessary to consider all of subordinate
skills.
Constructing a
learning hierarchy for a mathematical topic is more than merely listing the
steps in learning the rule or solving the problem. Preparing a list of steps is
a good starting point; however the distinguishing characteristic of a learning
hierarchy is an up-side-down tree diagram of subordinate and super-ordinate
abilities which can be demonstrated by students or measured by teachers.
Below
is the list of steps used to derive the quadratic formula.
And
then, the following figure is a first approximation to a learning hierarchy for
deriving the quadratic formula.
The figure above
is a learning hierarchy, because both super-ordinate and subordinate abilities
are specified in their appropriate relationship to each other. That figure can
be thought of as first approximation to the learning hierarchy for solving a
quadratic equation. A more careful consideration of prerequisite abilities and
research with students might result in a more precise hierarchy for this
problem-solving ability.
A.
Condition
of Learning
In order for learning to
take place certain conditions must be present. In 1965, Robert Gagne published his book entitled The
Conditions of Learning. In this book, Gagne described the analysis of learning objectives, and how they relate to the appropriate instructional designs
(Lawson, 1974). Gagne is considered to
be the primary researcher and contributor for the orderly approach to
instructional design and training. He is known as a
behaviorist and focused on the
outcomes or behaviors that result from instruction. The Conditions of Learning theory stipulates several different types or levels of learning. These classifications of learning are
significant because each type requires a different kind of
instruction.
Gagne distinguishes between two types of conditions, internal
and external. The internal
conditions are thought of as
“states” and include attention, motivation, and
recall (Wilson, 1978). The external conditions are
thought of as factors that surround one’s behavior, and include the arrangement and
timing of stimulus events
(Wilson, 1978). By taking these conditions into consideration, Gagne developed four phases for learning. These include
receiving the stimulus
situation, stage of acquisition, storage,
and retrieval. This is the
process that must occur for learning
to take place.
The Conditions of
Learning theory identifies
five categories of learning. Each
is a type of learning that can occur. They are verbal information, intellectual skills, cognitive strategies, motor skills, and attitudes. Gagne suggests that learning
tasks for intellectual skills can be organized based on their complexity. According to his model, the abilities established by
learning are
arranged in a hierarchical fashion where one task is dependent upon the learning of a more simplistic one
(Lawson, 1974). In order from simple to
complex are the learning tasks, stimulus recognition, response
generation, procedure following, use of terminology, discriminations, concept formation, rule application, and problem solving
(Lawson, 1974). The significance of the hierarchy is to identify prerequisites that should
be completed to assist learning at each
level and provide a basis for
the sequence of instruction.
After Gagne had identified the mental conditions for
learning, he created a nine-step process called the
events of instruction. This process includes the sequence of the instructional events
and the corresponding learning processes that guide the
instruction. The sequence for the instructional events
is gain attention, inform learners of objectives, stimulate recall of prior learning, present the content, provide learning guidance, elicit performance, provide feedback, assess performance, and enhance retention and transfer (Reyes, 1990).
By following these events the necessary conditions of learning should
be satisfied. These events
should also be the basis for designing
instruction and selecting the appropriate media for learning.
The nine-step instructional events
could be used in the teaching of classifying polynomials. By using each step students could
gain an understanding of what polynomials
are, what their purpose is, and how they can be applied to problem situations. The Conditions of Learning theory is quite
relevant to instructional use. It allows the instructor to first
understand how learning takes
place and the different types that occur. Then, it
assists in providing the type of instruction needed based on
that information. The most important thing to remember about
the conditions of learning is that
different instruction is required for different learning outcomes.
PREFERENCES
Bell, Frederick H. 1978. Teaching and Learning Mathematics (in Secondary Schools). Dubuque,
Iowa: Wm. C. Brown.
http://edtech2.boisestate.edu/fosterl/docs/504/504%20Paper%20(Conditions%20of%20Learning).pdf
By : Budi Santoso (083174016)
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