8

Transformation:

Schemas or Schemes

By

Joseph McNair

Our ability to change or alter our mental models and schemas to solve problems posed by new experiences or to understand what we are experiencing — what Piaget called accommodation — is the "mechanism " of personal transformation

In the preceding essay, mental representations, specifically propositional representations, mental images and mental models, were introduced and explained.  All of these are instrumental in facilitating or blocking the process of transformation.   The mental model, which subsumed (incorporated) the propositional representation and the mental image, is itself a component of more complex cognitive structures.  In this essay, we will examine one of these structures, the schema.

SCHEMAS
A schema, paraphrasing Piero Scaruffi (2003), is a network of ...[mental models]... that organize past experience.  The term schema may have been first used by the philosopher, Immanuel Kant.

http://www.mallorcaweb.net/masm/sissol/kant.GIF

According to Dan J. Stein (2003):

One of the greatest philosophers, Kant, employed the concept of the schema precisely in order to discuss the possibility of knowledge. Kant attempted to go beyond the impasse between the empiricists, who argued that knowledge has its origins in the external world, and the rationalists, who argued that knowledge is a product of the mind.  He argued that schemas interdigitate [To interlock, as the fingers of two hands that are joined] between properties of the mind (the a priori categories [derived by reasoning without reference to particular facts or experience]) and raw sensory data (the a posteriori experience [derived by reasoning from particular facts and experience to general principles]) "This representation of a universal procedure of the imagination in providing an image for a concept, I entitle the schema of the concept" (quoted in Gardner, 1985). Stein, D.J. (2003), Schemas  in  the  cognitive  and  clinical  sciences: An  integrative  construct [online][URL] http://cogprints.ecs.soton.ac.uk/archive/00001114/00/Schema.3.htm Schema.3.htm

Probably the most influential voice on the subject of schemes, especially within the developmental psychology and education circles is Jean Piaget.

http://www.ship.edu/~cgboeree/piaget.html

C. George Boeree (2003) gives a concise summary of Piaget's ideas about schemas and the processes of adaptation [learning]:

He...[Piaget]... noticed, for example, that even infants have certain skills in regard to objects in their environment.  These skills were certainly simple ones, sensori-motor skills, but they directed the way in which the infant explored his or her environment and so how they gained more knowledge of the world and more sophisticated exploratory skills.  These skills he called schemas.

For example, an infant knows how to grab his ...[or her] favorite rattle and thrust it into his...[or her] mouth.  [S/he's]... got that schema down pat.  When he ...[or she] comes across some other object -- say daddy's expensive watch, he...[or she] easily learns to transfer his...[her] "grab and thrust " schema to the new object.  This Piaget called assimilation, specifically assimilating a new object into an old schema.

When our infant comes across another object again -- say a beach ball — [s/he] will try his...[her] old schema of grab and thrust.  This of course works poorly with the new object.  So the schema will adapt to the new object:  Perhaps, in this example, "squeeze and drool " would be an appropriate title for the new schema.  This is called accommodation, specifically accommodating an old schema to a new object.

Assimilation and accommodation are the two sides of adaptation, Piaget's term for what most of us would call learning.  Piaget saw adaptation, however, as a good deal broader than the kind of learning that Behaviorists in the US were talking about.  He saw it as a fundamentally biological process.  Even one's grip has to accommodate to a stone, while clay is assimilated into our grip.  All living things adapt, even without a nervous system or brain.

Assimilation and accommodation work like pendulum swings at advancing our understanding of the world and our competency in it.  According to Piaget, they are directed at a balance between the structure of the mind and the environment, at a certain congruency between the two, that would indicate that you have a good (or at least good-enough) model of the universe.  This ideal state he calls equilibrium.

As he continued his investigation of children, he noted that there were periods where assimilation dominated, periods where accommodation dominated, and periods of relative equilibrium, and that these periods were similar among all the children he looked at in their nature and their timing.  And so he developed the idea of stages of cognitive development.  These constitute a lasting contribution to psychology. Boeree, C.G (2003), Personality Theories: Jean Piaget, [online][URL] http://www.ship.edu/~cgboeree/piaget.html

To illustrate how mental models are components of schemas it is necessary to discuss Piaget's theory in some detail. Piaget's stages of cognitive development referred to in the foregoing citation are

SENSORIMOTOR PERIOD
In the sensorimotor period, the predominant mode of learning used by the human being is the sensual exploration of the body and the environment through reflexive, then intentional motor activity. Most of our important motor (and mental) skills are developed from reflexes. Reflexes are triggered by sensation.  Sensory input -- a touch, for example -- is transformed into electrochemical impulses and transmitted through a sensory nerve to a junction with a motor nerve, where it goes straight to a muscle. The muscle moves!

Humberto Gutierrez and Christopher Ormsby (2003), at the Instituto de Fisiologia Celular, UNAM, in Mexico City provide a fairly understandable scientific description of the knee-jerk ( stretch or patellar) reflex.

http://www.dushkin.com/connectext/psy/ch02/reflex.gif

The knee jerk reflex is a well known example of stretch reflex. Tapping the kneecap (patella) pulls on the tendon of the quadriceps femoris, which is an extensor muscle that extends the lower leg. When the muscle stretches in response to the pull of the tendon, information regarding this change in the muscle is conveyed by the afferent sensory neurons to the spinal cord and the central nervous system. In the spinal cord the sensory neurons act directly on motor neurons that contract the quadriceps...The stretch reflex plays a central role in the maintenance of balance. Gutierrez , H. andOrmsby, C. 2003, Reflexes [online][URL] http://ifcsun1.ifisiol.unam.mx/Brain/reflex.htm

Reflexes in the newborn child such as sucking, rooting, grasping, the eye blink, the constriction/dilation of the pupil in the presence or removal of light, the Moro or startle reflex, the Babinski reflex,  and  the birth cry,  require no thought.

Sucking

http://js.dyck.org/images/stills/latest3.jpg

Rooting Reflex
http://www.avert.org/graphics/breastfeeding.jpg

Development Pattern: Disappears after 3 to 4 months\ 

Grasping
http://www.nctpregnancyandbabycare.com/nct-online/images/reflexes/palmar.jpg

   \\

   Blink Reflex\

http://www.nctpregnancyandbabycare.com/nct-online/images/reflexes/blink.jpg

 

        Dilation/Constriction of the Pupil

Adapted by author from http://www.physiol.cam.ac.uk/staff/lamb/Abstracts/ARVO/1998/2-eye.jpg

   

 Moro Reflex
http://www.nlm.nih.gov/medlineplus/ency/images/ency/fullsize/17269.jpg

 

             Babinski Reflex
http://www.tireswing.net/images/foot-01.jpg

 

     Birth Cry
http://www.siliconwoods.com/Joshua/Birth.jpg

These reflexes form the basis of later mental activity.   By becoming gradually aware of their reflexes, what their muscles are doing involuntarily, babies, first by "accident " and then through "practice " (repetition) begin to learn deliberate or intentional actions.

Instead of the rooting reflex—turning her head toward a stimulating finger or nipple to suck—the baby will learn with experience to combine the muscular activities of several reflexes to reach for and grasp a plastic feeding bottle and  suck on its nipple when she or he gets hungry.  Subsequently, through interacting with the environment and practice, reflex behaviors become intentional (and hence, thought about), organized, and complex.

Piaget suggests that these intentional behaviors, widely ranging in degrees of organization and complexity, are linked to structures of mental models and associated behaviors called schemas or schemes. 

As the child grows, gains experience, is taught (acculturated) by adults and peers and begins to develop personal strategies for functioning effectively in his or her world, reflex behaviors develop into cognitive schemas.

To paraphrase Glenn Doman (1974) who traces the development of reflexes into useful social/academic behaviors:

When the human being develops more complex motor skills, s/he suddenly "knows "  how to do things.  S/he "knows " how to walk, run, jump, throw, skip, dance, ride a bicycle, draw, play the piano. S/he learns gross and fine motor skills; s/he coordinates and integrates them through imitation and repetition. Once acquired, these skills require a minimum amount of conscious thought for competent performance.

Much of the adult behavior acquired before an individual truly understands what s/he is doing is based on the development of motor functions.  Motor functions, along with the ability to refer to mental models stored in the memory enable the individual to imitate certain behaviors so accurately that it often seems as if s/he knows what s/he is doing, especially if s/he achieves what might be considered a desired result.

During the sensorimotor period, the the following cognitive abilities emerge:

Basic Reflexes (Birth -1 month)

Children enter the world equipped with a set of inherited action patterns and reflexes through which they experience their environment. The intellectual development of the child begins through these actions as this is how the child acquires knowledge about its surroundings; this knowledge forms the basis for more complex developments further down the track. Infants are restricted in what they can know as their behaviours and schemata are limited. Adaption to their surroundings through assimilation and accommodation begins in this stage.

Primary Circular Reactions (1-4 months)

In the second substage of Piaget’s theory, the knowledge and intelligence of the infant now extends beyond the innate behaviors they were born with but these new acquistions have only come about through the accommodation of schemata. The infants show one of the first signs learning which is modifying their reflexes as a result of their environment (Bjorklund, 1995). These acquistions come about by a circular means. Actions that are at first random and activate a reflex are attempted again to try and induce the experience again. The signs of intentionality have appeared. These patterns of learning have been labelled primary circular reactions.

This is also the substage in which object permanence begins to develop and the active search for a hidden object begins.

Object permanence ..."refers to the ability of the brain to retain and utilise visual images. It develops at about eight months of age. This faculty is distinct from a baby's recognition memory. For example, a baby is able to recognise and prefers to look at its mother by the third day of life. However, it will not cry upon being left by mother; "Out of sight, out of mind." At around eight months, the child will be exhibit signs of separation anxiety when mother leaves the room. This is because the child can now appreciate what he has just lost - the presence of his mother. Another sign of the attainment of object permanence is baby's delight at the game of "peek-a-boo," which demonstrates graphically that the child appreciates that just because Mother is out of direct view she is still in the world and can be recalled by moving the hands or blanket out of the way. Stranger anxiety is another..."http://www.drhull.com/EncyMaster/O/object_permanence.html

Secondary Circular Reactions (4 – 8 months)

Secondary circular reactions are the first acquired adaptions of behaviors that are not reflexive, as opposed to the primary circular reactions which are reflex based. An infant in this stage may accidentally cause something interesting to happen and then seek to re-create the happy event. The interesting events in this case are located in the external wolrd, in primary circular reactions the interesting events are occuring within the body. A child in this substage, however, does still not understand the aspects of cause and effect and so will sift through the many behaviors it was indulging in when the event occurred and narrow it down to the particular action without really understanding the underlying concepts of why the event recurs.

Coordination of Secondary Circular Reactions (8-12)

The actions of the previous stage flourish in this stage and continue to develop, the difference is that the need now precedes the act. Intentionality occurs in interactions with the environment and the infant is moving towards goal directed behaviour. An understanding of cause and effect relationships has come into being in the childs world.

Tertiary Circular Reactions (12 – 18 months)

This stage is characterized by a means/ends differentiation. The infants are no longer restricted to the application of previously established schemata to obtain a goal. They can make the necessary alterations to their schemata to solve problems; this reflects a process of active experimentation (Bjorklund, 1995). These differences in cognition coincide with improved locomotive abilities; the children have become more physically active. In this stage, causal inferences are still unavailable to the infant; it must see an action occur before it has any understanding of the causal relationship.

While there is controversy on when object constancy emerges, several theorists believe that object constancy begins to emerge at the end of this substage.

Object constancy is .."usually defined as the capacity to see and relate to the other as a person in his or her own right. This capacity is part of the quality of being personal and able to make direct personal contact. Very young children child have a sense of life as being “All is me”. They have no sense of“I exist - and others also exist in the world being separate from me”. As a child develops object constancy it begins to realise that the object (mother) is separate from him. The problem with that is: “What if mother decides to go away” and thus he would shrivel up and die just like the baby bird who falls out of the nest and is left to die. By mother (the object) constantly being there most times the baby cries, then the child gets a sense of the object being constant. That is he starts to feel secure that she wont go away and secure that she and I are two separate people.Object constancy problems occur when the object is not constant. The primary parenting figure which is usually the mother for some reason is not in the infant’s life in a constant fashion. This can be obvious when there is clear neglect of a child. For instance the mother may have an ongoing alcohol addiction and thus the child is left for long periods of time with no care. Or there can be instances of benign neglect where the infant is adequately fed, clothed and housed but for some reason the emotional needs are neglected. For instance the mother may be ill, or have significant emotional problems of her own and so forth." http://www.ynot1.com.au/magazines/Object%20constancy.cwk%20(WP).pdf.

Invention of new Means through Mental Combinations (18 – 24 months)

The earlier stages of the sensorimotor period appear to be set on a continuum but the transition from the fifth to the sixth stage is more of a disjointed transition. Symbolic function and mental representation first appear during this stage, this runs parrallel with the development of language. Language is an expression of symbolic function and mental representation and it is at this stage that the children begin to string words together in pairs, the origins of sentences.

Piaget marks the beginning of "thinking " as occurring somewhere near the end of the sensorimotor period.  Thinking here refers to the ability to form, recall and manipulate mental images and simple mental models in the mind and use them to solve problems.  In other words, when the child is able to model the behavior of people, animals or animate objects without the "model " being present; and when the child begins to put together "cause " and "effect " relationships, that child has begun to think.

   Symbolic Thinking.jpg created by author

When an individual has an experience, s/he tries to fit that experience into an existing pattern of thought and learned behavior (a schema) and respond appropriately.  If the experience does not fit existing patterns of thought and behavior, s/he must either create a new schema or alter an existing one.

Experience, acculturation or social learning, maturation and disequilibration, a dissatisfaction with the efficacy of our current repertoire of schemas to solve problems inform the creation of new schemas or the alteration of existing ones.  Each new or altered scheme is an adapted and (hopefully) improved response to problems and/or challenges in the environment.  As problems in the environment become more complex, these schemas become more complex and more abstract.

PREOPERATIONAL PERIOD
According to Piaget, between the approximate ages of sixteen months or two years through six or seven years or what he calls the preoperational period. The child  uses the ability to symbolize—to form, recall and manipulate mental images and models — to represent the world symbolically.

Symbols are associated with perceptual experiences at a staggering rate. As the child develops language during this period, words and phrases come to symbolize mental images which represent perceptual experiences. A child's vocabulary increases from around ten words to over four thousand words over this four to five year period, enabling the child to begin to "think " in words or propositional representations.

At this stage children view all similar objects as though they are identical (Lefrancois, 1995). At this time all women are 'Mummy' and all men 'Daddy'. While at this level a child's thought is transductive. This means the child will make inferences from one specific to another (Carlson & Buskist, 1997). This leads to a child looking at the moon and reasoning; 'My ball is round, that thing there is round; therefore that thing is a ball' .

From the age of about 4 years until 7 most children go through the Intuitive period. This is characterized by egocentric, perception-dominated and intuitive thought which is prone to errors in classification (Lefrancois, 1995).

Most preoperational thinking is self-centred, or Egocentric. According to Piaget, a preoperational child has difficulty understanding life from any other perspective than his own. In this time, the child is very me, myself, and I oriented.

Egocentrism is very apparent in the relationship between two preschool children. Imagine two children are playing right next to each other, one playing with a colouring book and the other with a doll. They are talking to each other in sequence, but each child is completely oblivious to what the other is saying.

Egocentrism is very apparent in the relationship between two preschool children. Imagine two children are playing right next to each other, one playing with a colouring book and the other with a doll. They are talking to each other in sequence, but each child is completely oblivious to what the other is saying.

Julie: "I love my dolly, her name is Tina"

Carol: "I'm going to colour the sun yellow"

Julie: "She has long, curly hair like my auntie"

Carol: "Maybe I'll colour the trees yellow, too"

Julie: "I wonder what Tina's eyes are made of?"

Carol: "I lost my orange crayon"

Julie: " I know her eyes are made of glass."

These types of exchanges are called "collective monologues". This type of monologue demonstrates the egocentrism of children's thinking in this stage.

According to Piaget, egocentrism of the young child leads them to believe that everyone thinks as they do, and that the whole world shares their feelings and desires. This sense of oneness with the world leads to the child's assumptions of magic omnipotence. Not only is the world created for them, they can control it. This leads to the child believing that nature is alive, and controllable. This is a concept of egocentrism known as "animism", the most characteristic of egocentric thought.

Closely related to animism is artificialism, or the idea that natural phenomena are created by human beings. Such as the sun is created by a man with a match. "Realism" is the child's notion that their own perspective is objective and absolute. The child thinks from one perspective and regards this reality as absolute. Names, for example, are real to the child. The child can't realize that names are only verbal labels, or conceive the idea that they could have been given a different name.

During the pre-operational period, the child begins to develop the use of symbols (but can not manipulate them), and the child is able to use language and words to represent things not visible. Also, the pre-operational child begins to master conservation problems.

By the age of four children are developing a more complete understanding of concepts and tend to have stopped reasoning tranductively (Lefrancois, 1995). However their thought is dominated more by perception than logic. This is clearly illustrated by conservation experiments. In such an experiment a pre-operational child may be shown two balls of clay, that the child acknowledges are equal in size, one of which is then squashed. The child is now asked if both lots of clay are equal. A child at this stage will say they are not longer equal.

Although the child is still unable to think in a truly logical fashion, they may begin to treat objects as part of a group. The pre-operational child may have difficulty with classification. This is because, to a pre-operational child, the division of a parent class into subclasses destroys the parent group (Lefrancois, 1995). For example, a child has a pile of toy vehicles which are then split into trucks and cars. Next the child is asked 'Tell me, are there more trucks than vehicles, or less, or the same number?' the child will almost always say there are more trucks than vehicles!

In the latter part of the preoperational period, the child begins to have an understanding between reality and fantasy. The child also begins to understand sex roles in society.

CONCRETE OPERATIONAL PERIOD
From the approximate ages of six or seven years through twelve years or what Piaget describes  as the concrete operational period, the child develops the ability to distinguish "what is real " conceptually from "what appears to be real, " perceptually.  S/he establishes the basis for logic by learning to conserve—to hold in mind the fact that the length, area, mass, volume, or number of objects remain the same even though they may appear perceptually to be physically different. For example:

image taken from Psychology, Worchel, S and Shebilske, W. 1989, Prentiss Hall, p.319

A liter of water is still a liter of water whether it is in a short, wide beaker or a tall, thin beaker.

Pennies.jpg created by author

Ten pennies are still ten pennies whether lined up in a straight line or bunched together.

   http://www.science.mcmaster.ca/psychology/psych1a6/1aa3/Develop/dev2f.gif

A pound of clay is still a pound of clay whether shaped in a thick roundish lump or a long, thin "weenie, " etc.

Other logic-based tasks that children learn to perform in this period either from adults, through formal schooling, and especially through concrete experiences  are to:

  1. distinguish (define) objects by class or sets of characteristics;
  2. construct or identify rankings, ordering or hierarchies in and between classes of objects and phenomena
  3. add, subtract, multiply or divide objects belonging to different classes;
  4. remember the original "form " of a mental image through a transformation or series of transformations;
  5. determine whether or not an amount of substance changes when its shape changes;
  6. think ahead several sequences of simple behavior.

They also learn about rules concerning :

They learn about the consequences of adhering to or disregarding rules:

They learn how to conform. They learn the beliefs, values and attitudes of their adult caregivers, valued adult role models and their peers.  They do this by referring to schemas they already have to apply to problem situations, by adapting old schemas to solve new problems or creating new schemas. What they can't do very well or accurately in the concrete operational period, according to Piaget, is generalize, infer, hypothesize or abstract.

FORMAL OPERATIONS
The perod ranging between the ages of ten or eleven years and adulthood, which Piaget calls the formal operations period, is when children are expected  to "construct " complex  mental models and to generalize accurately—they can "image " in their minds the characteristics in common with phenomena and say declaratively "all so-in-so are  this or have that. "  For example:

Said twelve-year old Thomas, when asked what was an insect, "An insect is any  class of arthropods with a well-defined head, thorax and abdomen, only three pair of  legs  and one or two sets of wings. "

http://www.insect-sale.com/images/main.gif

During this period, children are expected (in western cultures) to learn to "logically " infer.  An inference is a conclusion derived from facts or assumptions taken to be true. To infer is to conclude or decide as a final logical result in a process of reasoning. For example:

Said thirteen-year old Marva,

"My mother is full-blooded Cherokee. My father is African American.  Therefore, I must be an African-Amerind. "

http://www.fattuesday.com/market/indian_posters/black_ea.jpg

Inferences are also drawn to explain behavior:

Why  do you  think  David  hit  you, Diane?
Cause he was mad.
How do you know he was mad?
'Cause of the way  he was looking at me,
like he wanted to jump on me.
Why do you think he was mad?
Maybe he didn't have no breakfast.

http://www.real-solutions.info/images/home/angry_boy.jpg

In this second example, we see that an inference is also "the discovery of one or more relationships between objects or events " (Sternberg, 1988, p. 117).

To hypothesize is to offer a tentative, testable explanation of a behavior, an event, a phenomena; the origins of or the cause of the same.  The operative word here is "tentative " which is derived from the Latin root "tentare " which means "to try or to touch. " The very use of the word tentative denotes trial or experiment. A conclusion derived by inference is an end point in a most western reasoning processes; a hypothesis is a beginning.

http://www.lifedesignsranch.com/mechanic.jpg

Arturo, 16, is ready to leave for school.  He gets into his car, puts his keys into the ignition and turns on the switch. Nothing happens. The first thing that comes to him is "it's probably the battery! "

He goes through several checks to see if the battery is working.  He tries the horn, it doesn't work.  He tries the radio, nothing.  He concludes that the problem does in fact lie in the battery. He then raises the hood of the car and checks the battery cables.

One of them is loose. He tightens the cable and tries to start the car again. This time it starts.

The thought, "it's probably the battery, " in the above example, is a hypothesis.   It is a tentative explanation, which requires testing. Whenever a hypothesis is given, a test of that explanation must follow. 

If, in the foregoing example, the radio and horn did work, and the cables were tight, Alvin would most probably reject the "battery " hypothesis and formulate another— "Maybe it's the alternator! "

Adolescents in what Piaget calls the formal operations stage of development can form rather complex hypotheses and test them "in their heads. " They can also solve problems whose forms  are the same even though their content is different.  An example of this is found in the solution of word-pair analogies, an important component of linguistic reasoning measured by intelligence or achievement tests. 

Sternberg (1988) illustrates the recognition and use of "forms " to solve problems in his explication of the steps in the solution of the following word-pair analogy problem:

WASHINGTON is to ONE as LINCOLN is to: (a. FIVE, b. TEN, c. FIFTEEN, d. FIFTY). (The correct answer is in bold-italic type).

To begin with, a person must encode, or think about, the various terms of the analogy, identifying each and retrieving from long-term memory the attributes that may be relevant to a solution. Examples of possible encodings for WASHINGTON are:

He was a President of the United States; he is portrayed on a piece of currency; he was a Revolutionary War hero. Some people fail to solve the analogy simply because they neglect (or are unable) to encode the proper attributes.  For example, those who do not encode either Washington or Lincoln as having his portrait on currency will be unable to solve this analogy.

Next, a person infers a relationship between the attributes of the first two terms of the analogy, WASHINGTON and ONE. Here, we may infer that Washington was the first President  or that he is portrayed on a one-dollar bill. Failure to infer that ONE can refer to the portrait of Washington on a one-dollar bill is a common error.

Now a person maps, or connects, the higher-order relationship that links the first half of the analogy, headed by WASHINGTON, to the second half, headed by LINCOLN.

Both halves deal with relationships originating with Presidents of the United States, portraits on currency, and war heroes. Failing to connect WASHINGTON with LINCOLN as portraits on currency leads to failure in the analogy solution.

Then a person applies the relationship inferred between the first two terms (WASHINGTON and  ONE) from the third analogy term, LINCOLN, to each of the possible answers.  Whereas Washington is portrayed on a one-dollar bill, Lincoln is portrayed on a five-dollar bill  . . .  (Sternberg 1988 p. 116).

http://www.treachery.net/~jdyson/dollar_bill_both_sides.jpg http://www.moneyfactory.com/uploads/five.jpg

The method outlined by Sternberg to solve the foregoing problem is a schema. It begins with

1.      the process described as encoding, then

2.      an inference of a relationship between the two parts of the analogy is made,

3.      the relationship  that links the first part of the analogy to the second is mapped, and finally,

4.      the relationship inferred between the first two terms from the third analogy term, to each of the possible answers is applied. 

When the student learns these problem-solving sequences, this schema, it can be applied to other problems that have the same form. The adolescent in the formal operations period of development, according to Piaget, should be able to recognize the "A is  to X  as B is to Y " form of a linguistic reasoning problem (analogy) and can apply the solution strategies e.g., encoding, inference, mapping and application, etc., to problems with the same form but with different contents. 

For example:

TRAP is to PART as RAT is to 

(a. GOOD-BYE,  b. WHOLE, c. BAIT, d. TAR)

This second problem differs from the WASHINGTON—LINCOLN problem in that the first requires selecting between the associations connected to the names Washington and Lincoln e.g., U.S. Presidents, portraits on currency, war heroes.  The second problem requires an examination of the words themselves or, more specifically, the order of the letters in the words.  Sternberg continues:

Since a reversal in the letters (encoding) of the word TRAP yields PART, the relationship (inference; mapping) that links the first part of the analogy to the second part is  the  reversal  of  letters  in the word.  TAR (application) can be the only correct solution.

Thus TRAP  is  to  PART as RAT is to TAR. (Sternberg 1988 p. 117).

The form (analogy) of the problem and general solution strategies are similar, but the contents of the problem and specific solution strategies are different (p. 317).  To be able to solve this and  other kinds of problems  mentally, the child must be able to manipulate fairly complex abstractions. There are several levels of abstraction. At the simplest level, there are nominal and body abstractions.  A simple abstraction is

the process of mental images or forming mental models of perceptual experiences related by common characteristics or common properties (adapted from Lefancois, 1980). 

A young child forms mental models through sensory experiences and creates simple schemas to solve problems posed by the environment.  Later, through the process of social interaction— "taking the word " or example of a parent, teacher or other cultural agents "—she learns to associate or "link " meaningful speech sounds (words) with sensory experiences.  She  learns to  identify the objects, people, animals  and events she experiences by the simple nouns and verbs.  She is told that the four legged animal that makes the "woof " or "bow-wow " sound, that has sharp teeth or licks her face and feels "furry " is a "dawg. "

Whenever she hears that particular speech sound, she can call up that mental model from memory. When she "sees " or experiences that particular animal, she can identify/recognize it because the mental model she has of it comes when retrieved with the speech sound "dawg. " She also knows what to do with or about the "dawg " She has blueprints for her behavior—schemas—which come with the mental model.

In the same way she learns to identify herself by name, to respond to her name when called, to identify/recognize herself by name in pictures and mirrors (self identification). It is crucial to the development of a healthy self-concept that a child is able to differentiate herself from objects and others in her environment and know herself by name. Shortly thereafter, the child learns to  name  and locate her eyes, hands, mouth, hair, nose, feet, eyebrows, fingernails, shoulders, elbows, knees, back, neck, chin, forehead, wrist, arms, legs, toes, etc., and describe them by their function.

The child learns that she is not the only inhabitant of the universe. There is herself and others  similar to her.  She has a name and they have names.  She has various body parts  and they do too.  When she can identify others by names and  pictures,  locate body parts on others, generalize to pictures, and complete body picture puzzles, she is abstracting. She has  formed  the concepts "me " (self) and "you/them " (others) (Valett, 1967).

Like simple abstraction, complex abstraction is based upon the mind's ability to associate, to link mental images and mental models together in the memory or in the imagination. Complex abstraction is often based on associations "further removed " from the simple "connections " of common characteristics or common properties.

http://216.219.160.214/gb/tense1/hands.gif
http://www.oceanoasis.org/teachersguide/images/a3-hands-2.jpg

In the example of body abstraction given above, the child learns that the body part she uses to pick up other objects, to point, to scratch her head is called a "hand. " As she explores her environment, she notices that her Mommy, Daddy, brothers, sisters and all of her friends have body parts that look like the ones she has which she has learned are called "hands. " She learns that these body parts are also called "hands " even though some are larger, smaller or a different color than hers.  Because they are shaped in the same way etc., and do the same things, she "abstracts " her own "hands " onto the bodies  of  others  around  her  and knows those body parts to be hands.

http://www.crafthome.com/images/presenttime/romanfacernd.jpg

A much more difficult abstraction for that child to make comes when she is shown a clock and  is told that the two (or three) stick-like "things " that move in a circle and point to the numbers on the clock are also called "hands. "  Why are they called "hands, " she might ask? She is told that they are called "hands " because they point to the numbers just like her own hands can point to things.  She must now associate the "things that point to the numbers " to the speech sound "hands. "

Another example of associations "further removed " in complex abstractions are number concepts and unknown quantities.  Let us take the number "2. "  The arabic numeral that we are familiar with is a symbol (an abstraction) of two things, any two things.

When we see

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3D_Apple_2.gif

 

we recognize them to be two apples.    When we see

 

adapted by author from http://www.twu.edu/o-ll/images/blocks.gif

we recognize them to be two blocks.

We recognize them because we have experienced them sensually and perceptually. They   are a part of our knowledge base or long-term memory and belong to separate classes of objects known as "apples " and "blocks. "

Apples are not the same as blocks, but in this case the quantity or number of apples and blocks are the same. Numerals (symbols of number quantities) evolved to enable us to represent quantities, not objects.  Therefore any time we want to represent the quantity  "two, "  regardless  of what  objects or things there are "two of " we use the numeral  "2. " 

The numeral "2 " represents any two things.

      

http://www.nottingham.ac.uk/education/number/2.gif
Roman Numeral II .jpg, created by author
Jewish letter number Beth (2).jpg ) taken from "Sacred Tarot ", C.C .Zain (1994)
Egyptian Letter/number Beinthin (2.jpg) taken from "Sacred Tarot ", C.C .Zain (1994)

Once we have learned how, it is easier and more efficient to mentally (or graphically) manipulate numerical symbols representing quantities of objects in the mathematical operations known as addition, subtraction, multiplication and division, etc., than to try to manipulate concepts of quantities and objects at the same time.

In the same way that we use "2 " to represent the quantity of any two objects or things, in algebra  and higher mathematics we learn to use the letter symbols  X, A, B, Y, etc., to represent any  number or  quantity. 

For example, the Commutative Law of Multiplication is stated in this fashion:

For any real numbers A and B (B can be any  number but A)

A B = BA

To test this mathematical law, we can substitute the number "1 " for A and the number "2 " for B:

1 x 2   =  2 x 1

1 x 2 = 2 and 2 x 1 = 2

2      =    2

It is neither practical nor possible for the average person to singularly test this law for "any number. "  But for it to be called a "law " one must suppose in the rationalistic tradition of mathematics that it has been tested a sufficient number of times for this property to be generalized to the entire class of "all real numbers. "

 

 

The adolescent in the Piaget's formal operations period is expected to "know " how to apply this law when, for example, simplifying equations.   S/he holds the "form "  [mental model] of this law as well as the experiences of applying the law [schemas] in symbolic mathematical operations in the memory and can retrieve it when confronted with mathematical problems which require its application.

Adolescents well into Piaget's formal operations period are expected to be able to understand and even create such language forms as similes, metaphors, proverbs,  sarcasm, satire and analogies.  The creation of these rather complex language forms suggests the ability to think in words and phrases, to combine and recombine linguistic symbols of people, objects, events and states of affairs with associated mental models to create new meaning, to create relationships where none before had existed—to construct knowledge.

Constructing knowledge is a socio-cultural process which begins with sensorimotor activity, perception, language development and social interaction.  Through these combined processes, we construct mental models and schemas.

Percepts, or meaningful sensory images, are mental symbols [images] which represent the objects, people and events of experience.  These are the building blocks for mental models.  How we assign meaning to mental images is largely a culturally determined process which relies heavily on the prescriptive authority of a given culture or subculture. 

As we continue to experience the world, we are actively fitting new percepts into existing mental models and schemas or creating new mental models and schemas.  We learn words to name concepts [mental models] and behaviors [schemas] and those words, mental models and schemas become symbols which we can manipulate mentally to solve problems. 

When we learn the rules of language and logic, we have schemas that allow us to:

1.      relate or connect unlike concepts, objects and events;

2.     apply concepts learned  from  particular situations to general situations;

3.      draw conclusions e.g., see consequences, make predictions from an examination of a set of facts or circumstances;

4.      offer temporary explanations for things that happen and test those explanations and

5.      make general statements about classes of phenomena.

With our mental models and schemas, we not only construct knowledge but our reality as well. Like our mental models and schemas, our reality begins as a simple construction that grows in complexity with new experiences and the knowledge that is distilled from those experiences.  As our reality becomes more complex, we can access higher and more structured levels of consciousness.

Our ability to change or alter our mental models and schemas to solve problems posed by new experiences or to understand what we are experiencing — what Piaget called accommodation — is the "mechanism " of personal transformation. When our models and schemes don't work, we must change them.  When we change these, we change ourselves. When we change ourselves, we change our reality.

Go to Essay 9:Transformation:Frameworks, Paradigms and Paradigm Shifts

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What can you do with an image? [online][URL] http://www.missouri.edu/~kingjw/psy240imagerylects.html