Learning Theory and Developmental Characteristics from
Infant - Adult
- Learning theory
- Learning cycle or procedure
- Learning variables- similarities & differences
- Explanations for misconceptions
- Physical activities to develop intelligence, logic, and thinking
- Developmentally appropriate
- Development by ages
Play is the highest form of research.
The most important factor influencing learning is what the learner already knows and that the job of the teacher is to ascertain this and teach accordingly.
All learning theories are heavily influenced by Jean Piaget's investigations of human development. Probably no other theory has undergone such scrutiny and verification than his ideas for the development of mathematico-logical reasoning. Particularly with respect to conservation and formal operational thinking. Questions with respect to the generalization of his ideas from research on his children to the world's population have been repeatedly investigated with virtually every culture on Earth with all attempts to find counter examples being generally unsuccessful. Further, his theory and subsequent studies have refuted Bruner's claim that anyone can learn anything at any age if it is presented to them in an appropriate instructional sequence.
Infants begin learning by storing perceptual memories of their sensory experiences while interacting with their environment through play, which begins at birth. Memories stored in different areas of their brains relative to how they are being sensed and how they are being acted on by the brain. A process which structures information as determined by the types of sensory input, previous memories, the brain's anatomy, and genetically programmed development. Development that enables our brains to make associations and construct better structures, actions and operations for thinking, understanding, and reasoning as we adapt to our environment, beginning as play and continuing throughout life as we notice and wonder.
This article describes the learning process, how people learn, develop, and use their intelligences, knowledge, and reasoning. It describes how reasoning can mislead children and adults. How over time it can develop to become more rational and logical; resulting in increased knowledge and intelligences for better understanding.
Children begin their construction of reasoning with direct observation and comparison of one observation to another. The consequences of reasoning with and about their perceptual senses creates a belief that what is being observed is reality, rather than a construction from the perceptual senses of external objects, which are represented as thoughts in our brains. The result, is not being aware that sensory information can be inaccurate. Over time children will learn how certain inaccurate sensory information can be corrected by operating on it with reason and logic.
Therefore, children are unaware which of their sensory perceptions are accurate or not and infer all are real. Acting on this information as facts and using reason to create concepts and generalizations from them. Operating on them with actions and structures that can lead to misunderstandings and misconceptions.
Reasoning can be conscious and unconscious, as our working memory loves peace and clarity, it will seek to maintain harmony with current understandings. However, as more information becomes available, conflicts arise, which can result in change if additional actions create new operations to use in different ways for other explanations and understanding. Operations, which may include: transformations, reversibility, conservation, perspective, classification, order, count, seriate, scale, proportion, relative position, motion, and other actions or operations of reasoning and understanding.
Examples of using perceptions as reality:
- Believing the moon is following them.
- Standing on a railroad track with an internal visual perception of the tracks and thinking the tracks are actually getting smaller and narrowing with distance.
- Thinking pennies are worth more than a dime because they are bigger, or two pennies are worth more than one dime because there are more pennies.
- Thinking five blocks grouped together is less than five blocks spread apart.
- Thinking one wire is shorter than another, because it's bent...
- Illusions and perceptual experiences that challenge reasoning
To correct inaccurate conclusions, learners engage in reasoning by using operations that act on their perceptions. Actions, which may be logical and accurate, but don't have to be. The learner just needs to believe it is. Examples:
- Railroad tracks are an equal distance apart, parallel. They don't get smaller, because when I walk down them they stay the same.
- A set of objects don't change their cardinality if they are moved, because when I count them and move them I get the same count. And besides they are just being moved no objects are being added or taken away. They are the same group.
- Seven is my lucky number, because when I roll dice I seem to get more sevens than other numbers. Although, sometimes I wish I would get a larger roll.
- Thirteen is unlucky, because I had a bad day on Friday the thirteenth.
- A crystal, stone, buckeye, rabbit's foot, ... will bring good fortune.
- A dream-catcher will keep nightmares away...
- An apple a day keeps the doctor away...
- An itchy nose means someone is thinking of you...
The use of different operations provides learners with experiences to construct and refine their mental actions and operations, such as those, which are needed to conserve.
- Decenter (center focus on one feature and ignore others),
- Use reversibility (able to do and undo actions),
- See transformations (able to see change as an infinite series),
- Stop the use of egocentric thinking, (recognize another point of view, activity in the world is external and perceptions are constructed internally) and
- Stop the use of transductive reasoning (replace faulty reasoning with logical reasoning).
Actions and operations used to explore and adapt to the world. Adaptability created when operations are connected, enlarged, and consolidated to create better structures and systems to better understand and navigate the world.
- Activities - directions, materials, & other infomraiton to explore mental operations and reasoning
- Discrepant events that challenge reasoning
Learning and development
Learning causes development through learning cycles. Learning processes which are repeated over and over again as a person matures and gains intelligences and adaptability as new knowledge is connected to previous understandings. Let's look review this process and its variables.
Let's take a close look at the learning process or cycle of learning.
It starts with a person playing or cruising through life, in equilibrium.
Remember a working memory loves consistency, clarity, and harmony.
An event happens that gives the working memory, a double take, cognitive dissonance. When something seems different or doesn't make sense and the person becomes, disequilibrated.
The person has two choices: they can ignore it, no learning, or focus on it and figure out what the heck is going on, learning.
Reflecting on the situation, accessing conscious and unconscious information the situation can be explored and acted on in different ways, with different actions and operations. If none fits with current understandings, or any makes more sense, than what is already know, then the information is remembered and connected to their previous information, assimilated.
However, if a new action, or way of operating on the information in the disequilibrating event, makes more sense, then a new action, operation, concept, generalization, structure, system ... is constructed, learned - accommodated.
Learned, not in isolation, but influenced by the situation of its creation. Which in turn influences how it will be remembered by the social, emotional, and cognitive sensory inputs at the time and how the brain responds to them, both consciously and unconsciously.
An essential part of a response is how the information is organized, which can be influenced consciously and unconsciously.
- Unconsciously, by emotional reactions, past experiences, and other subconscious responses.
- Consciously, by playing with ideas and seeing how they can be extended, applied, and changed. Testing the limits of the information for greater benefit. And recognizing benefits. Benefits of understanding the world, adapting to better ourselves, surviving, thriving, and developing of our self-efficacy to live a life worth living to improve the world.
As a person repeatedly experiences the world they cycle through learning cycles.
They learn, develop knowledge, and attain greater levels of intelligences or wisdom.
The process doesn't change, but what we learn and how we know does change.
It is these differences that cause people to claim, all people learn differently.
Which is true, with respects to what is attended to, what is learned, the time frame in which it is learned, and how information is connected or structured.
However, the fundamental learning process does not change.
The following chart identifies variables that effect growth and intelligence. And categorizes them as those that change and those that do not change.
It is the general unchanging learning process and knowledge of these variables that effect learning that provides educators methods and strategies to use as a powerful guides to facilitate learning. Outstanding educators are skilled at organizing learning environments by considering how variables, which do effect learning, are best manipulated in learning experiences, or tasks, to invoke a learning cycle that facilitates learning. See pedagogy.
However helpful knowing all this is, there still is significant variability that makes learning a personal activity. So personal that group instruction and mass schooling often fails for even the best educators and learners.
Knowing misconceptions is less about uncovering what learners don’t know than establishing what they do know so as to establish a place to begin to activate their thinking and provide opportunities to further deepen their understandings.
To discover different ways learners think it is helpful to review different ways their logic and ways of thinking move from less accurate to more accurate ways of understanding.
A variety of strategies, that are often illogical, to explain their world. Strategies based on their sensory experiences with their world based on their, age or maturation. Limited by the number and diversity of their experiences based on their current development of logic and reasoning at the time of each experience. A Constructivist Piagetian based learning theory is helpful in explaining this development. This section focuses on characteristics of some common errors related to reasoning and logic.
Children begin thinking with their sensory inputs (perceptions), which actual direct observations of reality. They do not know our perceptual senses construct representations of external objects or ideas to act and operate on with reason.
Eventually children begin to discover inconsistencies with their reasoning. Inconsistencies adults sometimes refer to as naive understandings or misconceptions.
Understandings, created by not having an adult's repertoire of actions, operations, conservation skills, classification skills, and logical reasoning to use to explain their experiences. Therefore, young children use internal sensory perceptions to directly reason with the objects and their properties they sense. Using associations to connect ideas that seem reasonable to them as explanations for understanding their experiences. Associations of comparison of one internal perception to another with an analogy to derive explanations, which can lead to illogical conclusions. Examples include:
- Animistic or Animism - speaking as if natural phenomena, living and non living things are alive or have a soul. Explanation as supernatural. God did it.
- Anthropomorphic or Anthropomorphism - speaking as if natural phenomena, living and non human living organisms have human motivations, characteristics, or behaviors. Explanation as a human reasons for doing it or similar act. The light quit working, because it was tired.
- Artificialism - speak as if a human, or conscious entity, created something not nature or is something natural. Thunder is bowling.
- Social - experiences in the world with family, friends, neighbors, TV, mass media, school, libraries, museums, shopping malls, are connected as explanations.
- Oral language - repeat phrases they hear as explanations without understanding.
- Written language - repeat ideas from what they have read as explanations
- Vocabulary - misuse of words as explanations without understanding.
- Metaphorically - use of a metaphor to explain. The sun is like a fire.
- Episodic - explanations that relate to an idea or to a previous event or a particular context, but are not explanatory or causal.
The more experiences a person has in exploring and explaining their world, the greater the information and operations each person has access to, to more accurately explain their world. All dependent on observational evidence processed with logical reasoning of their observations and organized to create reasonable explanations for which they can feel confident.
A person does not know from what one senses (observes).
When a person perceives, internalizes, and constructs knowledge; that construction requires action on objects and ideas. For young children those actions occur as play. To know an object, requires the child explore it and act on it. Actions that physical manipulate their world and the object within it; to learn about them, their properties, and any changes from their interactions. Additionally they learn how to think. How to construct and use actions, operations, reason, logic and other thinking skills to develop their intelligences.
Physical actions support concrete learning. During play, learners manipulate objects and are provided immediate consistent feedback to reinforce learning with and about physical objects discovered by sensory inputs. It is out there to discover, like the First Americans discovered America. Physical properties can be discovered. Physical properties such as rocks: hard, smooth, rough, hot, cold, and irregular shape which is hard to change. Physical properties of Mom: she is soft, warm, comforting, provides food, and comes when I cry.
Logic and mathematical knowledge is not discovered, it is invented. It does not exist before it is invented by the child. Numbers do not exist. They are invented, with the invention tied to the objects. Physical and logical knowledge is necessary to create the understanding of a number system. The idea that mom will come when I call. The idea that eight objects can be rearranged and still have a cardinality of eight. Both of these illustrates how the physical and logical-mathematical are tied together.
The physical knowledge of rocks can be discovered. They are separate objects that can be moved and maintain their shape, mass, color ... no matter in what position they are placed. Mathematical knowledge about them has to be invented. A number can be assigned to each object in a one-to-one correspondence and the final number assigned is eight no matter how the objects are moved (in one pocket or two) or which gets assigned which number.
Sufficient physical skill, and opportunities to play, provide children with the experience needed to invent all understanding: physical, emotional, social, logical, aesthetic, and any other classification of understandings. The following actions and uses of those actions in play activities are used to create their foundation for learning.
|Pushing||Slide, Roll, Jump, Skip, Walk, Run, Hop, Throw, Splash, Spill, Smash, Mash, Throw, Press, Scratch, Pluck, Shout, Blow||Water play, blocks, toys, sandbox|
|Pulling||Slide, Roll, Lift, Splash, Spill, Squeeze, Smash, Mash, Suck,||Water play, blocks, toys, sandbox|
|Balance||Hold, Drop, Stop, Still||Water play, stacking, toys, balance beam, swing, building activities|
|Balance and Push and Pull||Support, Carry, Pour, Wet, Water, Fill, Empty, Stir, Mix, Soak, Rip, Open, Dig, Shake,||Water play, balance beam, swing, sand box|
|Smell||Sniff, waft,||Food, flowers, perfume, spices|
|Listen||Hear||Music, stories, talk, video, plays|
|See||Observe||Pictures, video, real life, drawings|
|Talk||Shout, Cry, Giggle, Whine, Whisper, Hiccup||Stories, information giving|
Information that follows describes characteristics of children as they develop from birth to adult.
Developmentally appropriate practices as described by the National Association for the Education of Young Children.
Nine principles of developmentally appropriate practice
by the National Association for the Education of Young Children. 2020
- Development processes reflect an interplay between biology and environment.
- Domains of development both support and are supported by the others.
- Play promotes joyful learning.
- Variations due to cultural contexts, experiences, and individual differences matter.
- Children are active learners.
- Children’s motivation to learn is increased when their learning environment fosters their sense of belonging, purpose, and agency.
- Children learn in an integrated fashion that cuts across academic disciplines or subjects areas.
- Development and learning advance when children are challenged to achieve at a level just beyond their current mastery.
- Technology and interactive media can be valuable tools for supporting children’s development and learning.
Development by ages
- New borns can suck, grasp, cry, vocalize, move, and grasp to begin their exploration of the world. They will search for food, nipple, after first 20 minutes of life.
- Motion and exploration are random without intention or purpose. Can't differentiate motion of self and objects beyond them.
- New borns can recognize their mother's voice.
- Babies cries are different for different situations: hungry ("na, na" kind of cry is related to sucking), others for tired, ...
- New borns respond to facial expressions.
- Vision is blurred beyond 20 inches for first two months.
- Infants explore and respond with their physical senses (touch, taste, smell, hear, and sight). Can locate the source of sound and coordinate vision with manipulation.
- Will learn better if multiple senses are involved.
- Learn to differentiate objects and follow objects. When they are not sensing or experiencing an object the object is not represented in their mind. Thus adding meaning to phrases, out of site - out of mind. Objects that can't be seen, cease to exist, don't have permanence.
- Can differentiate between one, two, and three objects. And few or many objects.
- Four - eight months they will anticipate the position of moving objects and follow objects after they disappear from view.
- Object permanence - objects, beyond themselves, continue to exist even when they are not being observed (8-12 months) faces, sounds, objects. Before object permanency hiding a bottle doesn't initiate a search, just panic. When an infant has object permanence, they can reverse a bottle to get the nipple and they will search for an object when it is hidden. These changes cause peek-a-boo to progress from startling to fun to boring.
- Must believe objects have permanency, continue to exist, to learn about the world with the belief of what you discover and learn will continue to exist and have meaning. Meaning that is consistent so cause and effect, reason, and logic have value.
- Discover self. Recognize self in mirror at 18 months. Act as if the world revolves around them. May have a temper tantrum when the don't get what they want.
- However, they will comfort others when they are hurt or sick, eager to assist and help others, show empathy and share pain.
- Will babble to communicate and can understand three times more than they can communicate.
- Their skeletons harden, knee caps develop.
- At one year their height has doubled and weigh tripled.
- Twelve - twenty-four months. Object permanence leads to understanding of constancy of size, shape, and other physical properties of objects. This adds stability to mental representations (abstractions) making thinking about inferences for cause and effect about changes observed during exploration and experimentation of their environment.
- Shake rattle causes sound. Pushing and dragging on a touch screen changes the images or causes sounds. Crank the handle, music plays and Jack jumps out of the box.
- Understand movement not perceived, (can predict an objects path if they see it move toward a screen, disappear behind it, and know where it will emerge on other side) and can represent spatial relationships.
The brain interprets the world on a need to know basis; only adding additional information as curiosity and desire push for more exploration to attain more information and detail. The amygdala is involved in deciding how much to process and reacts to stress by processing less information.
- Learn language. Language is deferred imitation. It is never the source of physical or logical-mathematical knowledge with out reference to pre or concrete operations.
- Language can be an obstacle to the development of reasoning and intelligence. Use words without associating meaningful understanding.
- Language slows thought as thinking requires actions to think.
- Language is social arbitrary knowledge. Like physical knowledge it is verified externally, but not with physical knowledge (until children can read and write), but through people or social knowledge.
- Learn vocabulary with a one to one association.
- Simple classification of objects into unique one property categories.
- Can hold one property or variable constant while other variables change.
- Logic is simple one-to-one chained relationships or step by step, one step at a time thinking.
- Can use symbols or mental representations. Therefore, mentally being able to manipulate symbols and mental representations is - abstraction of thought. Symbolic play is when another object can be thought of as another thing or pers0n during play. Blocks become cars and guns. Teddy bears a child or friend. Symbols are entirely arbitrary and generally shared socially. However, they can be unique to an individual child.
- Creates mental images
- Use deferred imitation will repeat the sounds a person says trains and animals make, choo-choo, meow, moo, ... Will repeat actions they view. Play house, doctor, sports, ...
- Likes to Draw. Drawings will be flat 2 dimensional, with some limited detail, and proportion. Will draw what they know, available in their mental structures. Not what they see. Example of dragon and knights ...
- Verbal, likes to talk about everything and to everyone, nonstop.
- Use mutual exclusivity and perfect it with one-to-one correspondence. Big Bird example. Big bird is big bird. Can't have another name.
These characteristics are possible largely because of object permanence and being able to construct and manipulate mental representations of objects as they are sensed. Their representations and ability to manipulate or act on their abstractions are contained or limited through the use of the following operations or lack of use of its opposite, which must be constructed for concrete operations.
- Egocentric thinking, believe their view is the truth. Understanding is based on their sensory input. It is not a selfish point of view. They don't recognize that other people may have a different point of view or that another point of view could exist. Inability to differentiate between self and others. To think critically they need to recognize there are points of view other than their own. What I believe as reality is one way to perceive the world and other people might have a different view. Need to use critical thought, allocentric thought, or sociocentric thought, to differentiate between self and others. To develop, ask questions as to what do you think the other person saw? Why did you pick this one? Why did she pick that one? What did the character in the story think? What did another character in the story think? Why did they think different things? Why did they think the same? Intent is to develop the idea of different points of view.
- Reversibility is the inability to reverse a thought. All sensory motor operations, perceptions are basically irreversible. Can't undo senses, can't unsmell, untaste, unsee. Therefore, it can be understood that reversibility is counter intuitive and a mental construct that has to be invented. Like wishful thinking of sometimes wanting to undo time. Reversibility is the ability to work a problem in both directions. Use logic both ways. This is essential to understand the order of events in time. In mathematics the position of an object doesn't change the object or its cardinality as the object can be reversed, transformed, from one position to another and back again.
- Being able to move four objects from one position to another doesn't change the cardinality (the number of objects) as they can be reversed to the original position.
- Being able to imagine water being poured from a measuring cup into a tall narrow container and then from the tall narrow container to a short wide container will have the same amount, because pouring it back into the tall container would reverse the process and the amount would still be the same.
Being able to take two wires of equal length, bend one, set them side by side and mentally reverse the process of unbending the wire to its original length and declaring them to be equal in length.
- Knowing that if 3 + 4 = 7 then 7 - 4 = 3 is the reverse process. If you add four and get seven, then if you have seven and remove four the process has been reversed.
- If exercise causes you to breath faster, what could you do to breath slower?
If water makes ice how can it be reversed?
- If salt is added to water can it be reversed? If it can, can the salt be reversed again with more water?
- Water evaporation and condensation.
- Balancing activities like those in FOSS for second grade as well as balancing on an equal arm balance.
- Mixing paint and watching the color change. Can paint be made to go from light to dark to light again?
- What about light from a flashlight? If we slowly add slips of waxed paper what happens? What if we remove them one at a time?
- What about unrolling a ball of string? How long is it? If we roll it up and unroll it again how long will it be?
- If you are feeling pretty smug about your ability to use reversibility, then think about being in a market and looking at all the different shapes of containers and trying to identify which are equal or which has more or less. I bet the packaging companies know which containers look like they hold more than they actually do, because of our inability to use reversibility well.
- Uses centering focuses on a characteristic as an explanation or cause, that may or may not be accurate, without consideration of other options. Decentering is the ability to change ones thinking from one characteristic to another when considering an explanation or cause. Able to explore all or multiple variables of an action, operation, idea, or event. Eventually systematically and simultaneously.
- Does not recognize transformations. Transformation is understanding that change can involve an infinite progression of steps, usually incrementally small. However, a progression that becomes infinitely large would be included as well.
Transformation is a mental structure necessary to be able to conserve. Piaget investigated and wrote about it. In one study children were asked to draw the position of a pencil from the moment it left a table until it hit the ground. He found it took several years for children to construct the idea that the pencil was in an infinite number of positions from starting to fall till landing on the floor.
Transformations are important to understand a concept of variable. Being able to visualize how a variable is changed across a range of possibilities or is infinitely continuous, is necessary for understanding and explaining the world. Continuous as opposed to a series of steps, beginning, middle, end. Like Google map shows a trip with a list of turns rather than a continuous route for the journey. Or a number line as stepping stones across a stream rather than a continuous path across it.
Transformation also applies to moral issues of right and wrong. For example: when the pencil is in the hand (totally right) to the pencil on the floor (totally wrong). However, moral issues can also range from an extreme rightness to extreme wrongness with a continuum of exceptions in between. For example when is it okay to tell a lie? The extreme cases would be: never to always. However, situations such: Do you swear to tell the whole truth ....? Do you like my shirt? How was the dinner I cooked? Suggest the idea of transformation might a strategy to use in discussing moral issues.
Another example is science. Science takes the view that it can not be proved the pencil will hit the floor every time it is dropped. There are no 100% right answers. Everything is tentative. However, there are certain events or objects where we believe some facts or inferences (theories, laws) are closer to truth than others (99.9%). For example with the seasons. Scientist are pretty much in agreement that the Earth's distance from the Sun doesn't vary enough to cause the seasons. And that the directness of the rays does vary enough to cause the seasons. However, even those ideas are tentative and open to reevaluation if new evidence is discovered. Just because you never saw a purple cow, doesn't mean there is none.
While this idea is not too difficult for adults to understand it is difficult for young people as they are still constructing or assimilating a variety of ways to apply transformations. Then to compound the problem is what seems to be a human obsession of wanting to classify everything as right or wrong; black or white; ying or yang. Of course it isn't easy for teachers either when we are forced to evaluate tentative answers.
Related activities or tasks.
- Sequencing objects of different lengths (Piaget's famous dowel rod experiment)
- Drawing pictures of the animation of a pencil or other object falling from a table or hand to the ground. Image to the right shows the progression of development of this idea. Young children draw the pencil standing on the table and laying on the floor. They do not draw the pencil at an incremental position. Later, children will place the pencil between the starting position and its final resting place. However, they will draw more representations toward the final position than the initial position. Eventually they will recognize infinite possible positions and draw many Iillustration3) or label it as such (illustration 4).
- Animation by making flip books
- Mixing paint and watching the color change. Add a little bit more darker paint at a time to lighter or lighter to darker.
- Flying paper airplanes and see how slowly changing the weight (add one paper clip to the nose, fly it five times, measure the distance to the nearest yard or meter, record it, and add another paper clip and repeat till five clips. or fold the ends of the wings a little more at a time and see how it changes the flight.
- Uses transductive reasoning may or may not be accurate. It is usually based on what one has experienced, (concrete properties) and associating one idea from one object to another object or experience. Usually an association related to simultaneous actions without cause and effect or logical reasoning (deductive or inductive). Juxtaposition of reasoning. A leaf is thin and floats so thin objects float. A rock is heavy and sinks so heavy objects sink. Can result in magical thinking, animism, faking connections, and other faulty reason. Reasoning resources.
- Lacks conservation of reasoning - Children in the primary grades must be taught as if they are lacking conservation skills. Anything that involves measurement needs to be experienced in its actual size. If students are studying animals, then the animals need to be drawn, outlined, or traced on sheets of paper to the exact size of a typical animal in the species. When a height or length is referred to, then pieces of string, yarn, or adding machine tape can be used to mark and show the sizes concretely.
- Solar system models create similar complications. A bigger problem with a solar system model is scale. Learners don't begin to develop the abilities for scale until about fourth grade. Then when a solar system model is made the scale for the size of the planets as well as the distance from the Sun need to be similar for an accurate representation, which is complicated as demonstrated when calculating scales for the planet's diameters and distance from the Sun.
- Similarly for time. A timeline can to be drawn with times in increments of the students' lives or generations for their parents and grandparents.
- Drawings are mostly flat two dimensional that lack perspective and scale.
- Develop class inclusion - Use classification and generalization to solve problems (all dogs are animals, only some animals are dogs. A tree is not an animal, therefore it is not a dog. A collie is a dog, therefore it is an animal.)
- Develop serial ordering - Arrange a set of objects or data from smallest to largest or by other criteria and create a one-to-one correspondence. Arrange people from smallest to largest and and relate to age.
- Develop conservation ability to question perception and reason about reality. To use logic instead of perception to make determinations for what can appear obvious, but is not. See also conservation task directions, materials, & other notes.
- Conservation of length - Perception of two wires of equal length, a bent wire and a straight wire, suggests the straight wire is longer.
- Conservation of number - five is five no matter if it is five elephants or five mice. No matter if the five horses are gathered by the fence at the barn or if they are spread out across the whole 10 acre field.
- Conservation of liquid/volume - Perception of two drinking glasses, of equal volume, with one being tall and narrow and the second being short and stout, would visually suggests the taller has a greater volume. Abstract reasoning could cause us to represent the two by imagining the tall one shrinking in height as it increases in girth. However, this is hard to do. Teen campers, who after being taught how to judge their serving sizes of beverages were given short wide glasses and tall thin glasses, which held the same volume, to pour their morning juice. They poured an average of 9.7 ounces into the short glasses and 5.5 ounces into the tall glasses. When questioned, they responded they believed they had poured less than 7 ounces into the short glasses and more than 7.5 ounces into the tall glasses. The same experiment was done with 89 adults. Seventy-nine percent of those with short glasses underestimated how much they poured and seventeen percent with tall glasses underestimated. Similarly, bar tenders poured 31.3 percent more into a tumbler than the more slender highball glass: 2.1 ounces v. 1.6 ounces. While these are errors in conservation reasoning, they have big implications for overconsumption of sugary drinks, alcohol and self dosed medications.
- Conservation of solid/mass - a ball of clay will have the same amount of solid material or mass if it is rolled out as a snake, pounded as a pancake or rolled into a ball.
- Conservation of area - My favorite one to act out is a story about two farmers: Mr. Smith and Mrs. Jones. Who have a farm yard with the same amount of grass.
- Mr. Smith and Mrs. Jones. Both have a farm yard with the same amount of grass.
- Place two identical pieces of green construction pieces.
- Ask, Which farm yard has more grass? (neither they are the same.)
- Okay good. Now on with the story.
- Mrs. Jones decides one day that it is time for her farm to grow so she builds a barn ( I place one Monopoly hotel on the green piece of construction paper).
- Mr. Smith, wanting to keep up with the Jones, build a barn exactly the same size ( I place one Monopoly hotel on the other green piece of construction paper).
- The next year Mrs. Smith decides that things are even better and builds two more barns along side of the first barn ( I place two more Monopoly hotels on the green piece of construction paper right beside the first).
- Mr. Smith, again wanting to keep up with the Jones, builds two more barns, but spreads them out across the barn yard ( I place two more Monopoly hotels on the green piece of construction paper so that all three are spread across the page).
- The following year Mrs. Jones has another good year and builds two more barns again side by side her original ( I place two more Monopoly hotel on the green piece of construction paper beside the other three).
- Again Mr. Smith, keeping up with the Jones, builds two more barns only he decides again to build his barns spread out across the farm yard ( I place two more Monopoly hotel on the green piece of construction paper spread apart from each other as much as possible).
- If there are more hotels, I continue until all of them are placed on the green construction paper, making sure to add them equally.
- Then one day Mrs. Jones goes out and buys a cow.
- Mr. Smith, wanting to keep up with the Jones, goes out and buys a cow.
- When both cows are let out into the farm yards, which cow is going to have more grass to eat?
- Most people who have never heard this problem before will have as their first hunch or answer the cow in Mrs. Jone's pasture. Just look at all that green. The image of the red (barns) hotels spread across the green paper and image of the green paper with barns huddled together in a corner can create a preoperational non conservation of area response or Mrs. Jone's cow based on the visual evidence. However, if you are skeptical enough, or whatever, to ask yourself to just wait and think here. You might reason that the amount of green covered by the same number of barn/hotels is equal, therefore, each cow would have the same amount of grass. If you are still not sure, act it out. Or get two pieces of graph paper and pencil in two squares for each barn. On the one sheet color them all in a corner and on the other spread them out. Are they the same area? Is the amount of grass the same? In order to conserve area a person needs to be able to stop their egocentric thinking and reason: Maybe there is a different way to think of this, rather than just looking at the grass for each, decenter (how else can I think about this problem rather than by looking at the amount of green and comparing), transformation (think what would happen if all the spread out barns/hotels started together and were moved/transformed to where they are now) or (what would happen if all the grouped together barns/hotels were moved/transformed to where they are now). It wouldn't matter where they are as they would be still sitting on the same amount of grass. Reversibility - If I can move the barns, they can be spread out, and then, they can be returned rejoined. Egocentric thinking is replaced by logical-mathematic reasoning.
- Develop use of perspective in drawing. Can begin to draw representations of three dimensions in two dimensional sketches and will draw with more accurate scaling. About second or third grade students are fascinated by following a procedure to draw a three dimensional representation of a cube on paper. About fourth grade, students are fascinated with the use of perspective and vanishing points to show perspective in their drawings. However, consistent systematic use will not be maintained until later in a formal operational sense.
- Develop the use of operations - Being able to use these operations enables people to make direct references to familiar objects and actions and explain them with associations. Example garduckals. Using associations allows them to create simple hypotheses, follow step by step directions or explanations, and understand others may have a different point of view or use a different reference point than they have or are using at the present.
- Because concrete operational learners begin with their own concrete experiences and associations, rather than with a formal operation, their abstractions are not as comprehensive and flexible with respect to generalizing less familiar or unfamiliar situations. Hence:
Concrete operational learners can identify some variables that interact with an object or system, but do not use a systematic process to identify them, usually resulting with an insufficient list of variables, and do not consistently plan and hold variables constant that are not manipulated or responding variables.
- Knows the difference of observation
and inference and can make inferences from observations, but considers a limited amount of possible inferences for an observation.
May apply a related, but inaccurate procedure or algorithm as a solution to a problem.
Apply a process without applying his or her own validation of the application of the process against the data, conclusions, or whatever information the situation provides for checking and validating a particular method.
- Learners will explore operations and procedures, which are usually indicative of formal operational thought, such as: isolation and controlling of variables, hypothesis, combinations, probability, correlation, proportion, and formal logical reasoning. However, they will do so with concrete objects or drawings that can represent the actual objects being considered and the operations that are being imposed on those objects. It is through these experiences learners are able to think about the operations and with enough experience begin to systematize them so they can become fluent enough to apply them systematically as abstractions. The development of fluent operations, for most learners is well after 12 years of age. See chart below for developmental ages of propositional logic, correlation, probability, and proportion.
Age 11 - adult, sixth grade and up Developmental Characteristics
How does formal operational thinking fit with the way adults think?
The characteristics of formal operational thought is not just abstract reasoning. Young children (preoperational) use abstractions to connect symbols or other representations to think and reason about real objects or events in their lives.
Formal operational thinking is being able to use abstract reasoning for operations which include more than one direct relationship. It requires reasoning with multiple properties or variables simultaneously, such as propositional logic, correlation, probability, and proportion.
Formal operations implies being able to understand a mental operation or procedure so well that it can be performed on information in an efficient systematical way. Operations and procedures that can be indicative of formal operational thought include isolation and controlling of variables, hypothesis, combinations, probability, correlation, proportion, and formal logical reasoning, which becomes possible at about 11 years old or sixth grade.
Being formal operational for the adolescent means, for the first time in his or her life, they have the mental capacity to think as well as adults and the ability to solve all classes of problems.
While formal operational thinking requires time for the brain to develop; time alone is not sufficient to guarantee formal operational thinking will develop. And one should not assume all adolescents and adults have fully developed formal operations. In fact a majority of adults never advance beyond concrete operational reasoning. See graph with four types of reasoning and scores across age.
While formal operational thinking provides ways of thinking about problems and information comparable to any advanced adult's way of thinking about similar problems; it doesn't mean the number of experiences a person has engaged in during their life time doesn't make a significant difference in their ability and efficiency of solving problems with or without formal operational thinking.
Additionally one may have the ability to use formal operational thinking in one or more particular areas and still not be able to generalize or transfer that formal operational knowledge to other areas. Hence, they rely on concrete operational thinking in those areas.
What does it mean to be formal operational?
Piaget claimed that after the development of formal operations any gain in a person's reasoning abilities is with respect to a person's ability and experiences with the use of logical operations and the efficacy of the individual's mental structures constructed with logical operations and the number of meaningful experiences the individual can associate with the use of the specific logical operations or combinations of those operations.
In other words, there is no higher level of reasoning beyond formal operational thinking. Differences in reasoning among formal operational thinkers is based on their mental power of short term memory, construction of accurate logical procedures, ability to mentally manipulate information from one form to another using an appropriate procedure, and creativity or flexibility achieved by experience to recognize what procedure fits in a particular situation.
Formal thought and concrete thought are similar in that they both use logical operations. However, there is a clear difference in the greater range of reasoning with the type of logical operations used at a higher level of understanding, described as formal operational thinking. A level of understanding, described as concrete thinking lacks systematic analysis, depth and range of comprehensive power, imagination, and flexibility of reasoning. In addition a formal operational thinker is aware logically derived conclusions have a validity different than conclusions directly derived from only facts and observations.
Formal operational thinking characteristics:
- Work with complex verbal propositional reasoning that is not tied to a personal past or present experience.
- Reasoning about hypothetical problems - reasoning that is not tied to a personal past or present experience and can project into the future without being tied to a personal past or present experience.
- Use theories, models, and hypotheses to create solutions to problems. Hypothetical reasoning goes beyond the confines of everyday experience to things for which we have no experience. Reasoning beyond perception and memory about things which we have no direct knowledge. Young adolescents with formal operations can reason about hypothetical problems entirely symbolically in their minds and can deduce logical conclusions.
- Think about his or her own thoughts and feelings (metacognition) as if they were objects.
- Reasoning can be independent of content. Can argue on the logic of an argument (solution or problem) independent of its content.
- Complex problems can be dealt with simultaneously and systematically by coordinating multiple thinking and reasoning strategies and or variables to derive solutions.
- Use inductive reasoning by combining similar solutions to create generalizations, principles, models, and theories.
- Have a highly developed understanding of causation.
- Use deductive reasoning. The use of a premise to create conclusions or the use of general ideas to create specific ideas. Inferences or conclusions created with deductive reasoning are true only if the premises used to create them are true. However, reasoning can use false premises and create logical conclusions.
- Use hypothetical deductive reasoning or reasoning with the use of a hypothetical premises (rather than facts) to create conclusions.
- Combinatorial reasoning is thinking that systematically considers all possible relations of experimental or theoretical conditions, even though some may not be realistic.
- Identify and control all variables when attempting to validate a relationship or inference. Designs a test that controls all variables, but the one being investigated.
- Use proportional reasoning
- Use probabilistic reasoning
- Use correlation reasoning to recognize a comparison between the number of confirming and disconfirming cases of a hypothesized relationship to the total number of cases.
These understandings combine to enable an individual to accept an hypothesized statement or assumption as a starting point for reasoning about a situation. He or she is able to reason hypothetic-deductively. OR... Able to imagine all possible relationships between the variables, deduce the consequences of those relationships, then empirically verify which of those consequences, in fact occurs. Daniella in the transportation puzzle demonstrates this.
Sample questions to apply and discuss what you know about human development
Discuss how each problem might or might not be solved at the different levels: sensorimotor, preoperational, concrete, formal operational.
- A < B; B < C what is A compared to C?
- Chris is left of Sam and Sam is left of Ben, Where is Chris in relationship to Ben?
- Formal operational can solve this problem in this form.
- Concrete operational can not. However, they can if they create the relationship concretely, draw and label pictures.
- Suppose snow is black?
- Young children will respond with, snow isn't black and possible not want to entertain an argument based on a false premise.
- Older children might think a false premise can be taken as a hypothetical situation that is interesting to speculate about and come to logical conclusions that can be inferred from the hypothetical situation.
- Formal operational, can reason logically or analyze the structure of an argument, independent of the truth or falseness of its content. How does the definition of formal operation relate to the two responses?
- Understand relationships between multiple variables simultaneously. Given an equal arm balance constructed so that the weights can be hung at equal increments from the center if three weights of the same mass are placed six units from the center how many weights of equal mass have to be placed three units on the opposite side to balance?
- Control multiple variables. Students are given the following equipment and asked to investigate and find what effect the kind of material, the thickness of the material, and the length of the material have on the flexibility of the material. Equipment: thick rods of aluminum and wood, medium rods of aluminum, steel, and wood, thin rods of wood and steel. All rods are the same length about 25 cm.
- Understand correlation. Example: What is the correlation of the number with blonde hair and blue eyes and the number with brunette hair and brown eyes?
- A random sample of 100 people is selected. They are surveyed, counted, and classified. Information resulted in the number with blonde hair and blue eyes; the number with brunette hair and brown eyes; the number with brown eyes and blond hair; the number with blue eyes and brown hair. The non-matching pairs were subtracted from the matching pairs and then compared to the total number in the sample. If the comparison is by division, then the result will be from 0 to 1 for a positive correlation and from a -1 to 0 for a negative correlation. With +1 resulting from a sample where all have blonde/blue or brown/brown matches or the entire sample correlates with the hypothesis. With a -1 resulting from a sample where all have blonde/brown or brown/blue matches, or the entire sample doesn't correlate with the hypothesis. Or with 50 50 resulting in a correlation of zero or no correlation with the hypothesis.).
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