Reversibility refers to the ability to mentally reverse or undo an action, process, or operation. This means that a child can mentally reverse the sequence of steps of an observed physical process.
Reversibility is a concept from Piaget’s theory of cognitive development. According to Piaget, children develop reversibility during the concrete operational stage, which occurs between the ages of 7 and 12.
Reversibility in Piaget’s Psychology
Inhelder and Piaget (1958) themselves provide a definition of reversibility:
“Reversibility is defined as the permanent possibility of returning to the starting point of the operation in question” (p. 272).
A few pages later they provide a more abstract example:
“Thus the reversibility characteristic of concrete systems of relations consists of reciprocity. For example, a symmetrical relation such as A = B is identical with its reciprocal B = A. For asymmetrical relations, if A < B is true, its reciprocal B < A is false;” (p. 274).
For example, in the famous conservation of liquid task, the child is shown two glasses that that are shaped the same and have the same amount of liquid.
The child then watches as the contents of one glass is poured into a taller and thinner glass.
Before children can perform mental reversibility, they will say the taller glass has more than the other.
But, when children can mentally reverse the pouring of the liquid back into the shorter glass, they will say the taller glass has the same amount as the shorter one.
At What Age and Developmental Stage do Children Develop Reversibility?
Most children will develop reversibility in Piaget’s concrete operational stage, which lasts from ages 7-11 years old.
Children in this stage develop increasingly advanced reasoning. Their thinking becomes better organized, more logical and methodical.
They can apply the rules of logic to physical objects and represent those objects mentally.
However, children still struggle with abstract reasoning.
The concrete operational stage serves as a transition between the preoperational and formal operational stages. Although the sequence of stages is invariant, children go through the stages at different rates.
- The Balls of Clay Task: A child sees two balls of clay that are exactly the same shape and mass. One ball is flattened. If a child can perform reversibility, then they will say that both still represent the same amount of clay.
- Pouring Juice Back into the Carton: For children that are able to perform reversibility, they will understand that if they poured too much juice into their glass, it is possible to pour some back into the carton.
- The Number Task: A child sees two rows of buttons. Each row has 5 buttons equally spaced apart. One row is spread out a bit. If a child is capable of reversibility, then they will recognize that each row still has the same number of buttons. The child will mentally push the buttons back to their original positions, which shows that each row has the same number. This video demonstrates this task starting at 0:38.
- Being Able to Fix a Broken Toy: When a very young child breaks a toy, such as pulling a wheel off a truck, they will immediately take it to the nearest adult to be fixed. As children get older however, they understand that they can fix the toy themselves by reversing the sequence that occurred when it was broken.
- The Length Task: A child sees two equal length sticks on a table. They are placed in parallel next to each other so the ends match up. Then one stick is moved to the left so that the ends don’t match up anymore. A child that can mentally reverse the movement of that stick will say the sticks are equal. But, a child that cannot perform this mental reversal will say that one stick is longer than the other. This video demonstrates this task starting at 1:18.
- Reversible Arithmetic: Reversible operations exist in algebraic equations and simple arithmetic. For example, teaching a child that 2 + 3 = 5 can be reversed to 5 – 3 = 2 may seem simple enough for an adult, but can be a real challenge for children just beginning to acquire reversibility thinking.
- Conservation of Area Task: Show a child two sets of 6 squares of green paper, each set arranged the same way. Next to each set is a photo of a cow. Explain that the cow wants to eat the grass (green squares). If you spread one set of squares out a bit, a child that can’t perform mental reversibility will think that cow has more grass to eat. However, if the child can reverse the spreading out action, they will understand that the total amount of grass has not changed at all. This video demonstrates a different way to test conservation of area.
- Playing with Ice Cubes: One activity to help kids understand reversibility involves letting them observe ice cubes melt in a glass over time. Then, let the child pour the water into an ice cube tray and put it back into the freezer. This will let them see reversibility using a concrete example.
- Understanding Correlation and Causation: Being told that factor X is correlated, but does not cause factor Y, requires reversibility thinking. A person needs to think about the direction of the causal relationship and that factor X must precede factor Y. The inability to engage in reversibility will not allow for that analysis.
- The Weight Task: Place two balls of playdough of equal mass and shape on a scale. Point out that the scale is balanced. Then, take one ball off and flatten in. If a child can perform reversibility thinking, then they will say that the scale will still be balanced if you put the flattened ball on the scale.
Case Studies of Reversibility
1. Reversibility And IQ
According to Marwaha et al. (2017), progressing through Piaget’s cognitive stages should be directly related to children’s IQ.
Marwaha et al. (2017) examined the relation between IQ and three cognitive abilities: centration, egocentrism, and reversibility.
Participants included 300 children in Delhi, India. The researchers focused on ages 4-7 in an attempt to identify the emergence of the three cognitive abilities.
IQ was assessed using the Seguin form board test.
To assess reversibility, two clay worms of equal lengths and amount were presented to the child. After the child indicated that the worms had the same amount of clay, one was changed into a wiggly shape.
The child was asked again if both worms still had the same amount of clay.
If the child answered in the affirmative, it indicated that they were able to reverse the change of the wiggly worm back into its initial shape.
Nearly all of the children failed to demonstrate reversibility. However, in regards to centration and reversibility, the researchers state that:
“There was a gradual reduction in the prevalence of these characteristics with increase in age from 4 to 7 years” (p. 115).
2. Understanding Family Relations
Teaching children about relationships between relatives is a challenge for children that are not yet capable of reversibility thinking.
For example, asking a child if they have an uncle might be easy for them to answer. But, if the question is phrased in reverse, it becomes much more difficult.
So, asking the same child if their uncle has a niece/nephew will often lead to a puzzled look, or maybe even a wrong answer.
This video shows a 9-year-old that can perform reversibility, but it takes a little bit of thinking.
However, the younger child in this video has not yet acquired reversibility thinking and so gives an incorrect answer.
If we were to examine this latter video from a more critical point of view, we might be able to generate some alternative explanations for the child’s error. Any ideas?
3. Teaching About Farm to Table
There are many ways to help children develop the ability to perform reversible thinking. This has been a particular focus in teaching mathematics and principles of thermodynamics because students often have great difficulties in these subjects.
However, reversibility is a valuable concept that can also be applied to subjects such as agriculture, sustainability, and food appreciation.
It is important for children to understand where their food comes from.
Instead of just seeing it magically appear on their plate three times a day, helping them understand the process of how that happens will instill a greater appreciation for the environment and sustainable practices.
This is why many schools have children engage in Farm-to-Table projects, so that they can see the process behind the end result. That often starts with a short video that explains the process.
This is a form of reversibility thinking that may not help a child solve an algebraic equation, but it will help them develop reversibility thinking in an environmental context.
4. Reversibility in Computer Programmers
Computer programming requires advanced reasoning skills and the ability to think abstractly, an attribute that is not common among all computer science students.
As Kramer (2007) asks:
“What is it that makes the good students so able? What is lacking in the weaker ones? Is it some aspect of intelligence? I believe that the key lies in abstraction: the ability to perform abstract thinking and to exhibit abstraction skills” (p. 37).
Lister identifies reversibility as one of the key abstract reasoning abilities. He illustrates the point with a simple coding problem that requires the programmer to reverse the operation of an array.
The novice programmer without reversibility might apply “random code changes and copious trial runs” (p. 15).
In contrast, a skilled novice will solve the problem by applying reversibility. There is no need for trial-and-error changes to the program.
Instead, the novice with reversibility, “after inspecting the given code, produced a correct solution almost immediately” (p. 16).
Lister recognizes that professors tend to “talk about programs in terms of formal operational reasoning.” But many students “tend to reason in a preoperational form” (p. 17).
The solution? To develop “a pedagogical approach informed by neo-Piagetian theory” (p. 17).
Reversibility thinking is the ability to mentally reverse operations or a process. It is one of the more challenging cognitive abilities to develop, but it is essential to mathematics and advanced subjects such as physics, thermodynamics, and computer programming.
Piaget and others have developed several ways of assessing reversibility thinking in young children. The procedures basically involve presenting the child with a set of equal objects, and then transforming one.
Children that have the ability to perform reversibility can mentally rewind the process of transformation and see that the transformed object can be returned to its original form.
Some educators have lamented the difficulty of teaching students reversibility thinking in more advanced subjects such as programming. This kind of abstract reasoning may simply be beyond the abilities of a certain percentage of students.
Inhelder, B., and Piaget, J. (1958). The growth of logical thinking: From childhood to adolescence (A. Parsons and S. Milgram, Trans.). NY NY: Basic Books. (Original Work published in 1955).
Kramer, J. (2007). Is abstraction the key to computing? Communications of the ACM, 50(4), 36-42.
Lister, R. (2011, December). Concrete and other neo-Piagetian forms of reasoning in the novice programmer. In Conferences in research and practice in information technology series.
Marwaha, S., Goswami, M., & Vashist, B. (2017). Prevalence of Principles of Piaget’s Theory Among 4-7-year-old Children and their Correlation with IQ. Journal of Clinical and Diagnostic Research : JCDR, 11(8) doi: https://doi.org/10.7860/JCDR/2017/28435.10513
Piaget, J. (1952). The child’s conception of number. London: Routledge & Kegan Paul Ltd.
Piaget, J. (1959). The language and thought of the child: Selected works vol. 5. Routledge, London.
Piaget, J. (1964). Part I: Cognitive development in children: Piaget development and learning. Journal of Research in Science Teaching, 2, 176-186. https://doi.org/doi:10.1002/tea.3660020306
Watanabe, N. (2017). Acquiring Piaget’s conservation concept of numbers, lengths, and liquids as ordinary play. Journal of Educational and Developmental Psychology, 7(1). 210-217. https://doi.org/10.5539/jedp.v7n1p210