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.
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. Students 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 lacking perspective and scale.
- 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.
Age 8 - 11, grades 3 - 5 Developmental Characteristics
- 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. Conservation tasks
- 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 might cause us to represent the two by imagining the tall one shrinking in height as it increased in girth.
- 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 and 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. Can you see that in order to conserve area a person needs to be able to stop their egocentric thinking (maybe someone else would think of this differently than just by 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 were they are still sitting on the same about of grass. Reversibility - If I can move the barns to spread them out, then they can be returned to being together. 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 dimensions representation of a cube on paper. About four 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 that they have or are using at the present.
- Because concrete operational students begin with their concrete experiences and associations, rather than with a formal operation their abstractions are not as comprehensive and flexible with respects to generalizing less familiar or unfamiliar situations. Hence:
Concrete operational students 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.
- Students will explore operations and procedures 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 students are able to think about the operations and with enough experience can begin to systematize them so they can become fluent enough to apply with them systematically as abstractions. The development of these operations not becoming fluent, for most students until 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 he or she has 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 that all adolescents and adults fully develop formal operations. In fact a majority of adults never advance beyond concrete operational reasoning. See chart above.
While formal operational thinking provides ways of thinking about problems and information comparable to any advanced adult's way of thinking about the same problem; it doesn't mean the number of experiences a person has engaged in during his or her life time doesn't make a significant difference in his or her 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 their formal operational knowledge to other areas. Hence, they must 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 specific logical operation or combination of 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 construction of accurate logical procedures, the ability to mentally manipulate information from one form to another using an appropriate procedure, and the creativity or flexibility, achieved by experience to recognize what procedure fits with 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 correlational 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.).