I have one that's not conventional, but is fun nonetheless. I remembered this experiment as we did it last year as part of school in 1/2 psych and I thought of sharing it! It's fun to do on a younger child, perhaps a sibling or cousin. It's based on Jean Piaget's model of cognitive development. This theory has since been rejected as it has been supported that children don't move through these stages chronologically and at the exact age timeframes, and the theory also assumes that children have developed language abilities to a high level by a young age. Basically, Piaget proposed that all individuals go through 4 stages chronologically: sensorimotor stage (BIRTH TO 2 YEARS OLD), preoperational stage (2 TO 4), concrete operational stage (7 TO 11) and formal operational stage (BEGINNING AT AGES 11 TO 15). You can read about each stage
here.
Basically, test the child on each stage's key behaviours chronologically and see up to what stage the child can go up to correctly.
Sensorimotor-
Object permanence (knowing an object exists even though they cannot physically see it): Show the child a toy, then put it under a blanket where the child can see you doing this. Ask them where the toy has gone.
Preoperational-
Egocentrism (tendency to perceive world solely from one's own point of view. Thinks others see the world in the same way that they do): Take a piece of paper that is not transparent when held up. Draw a flower on one side, and a smiley face on the other (or any pictures that suit your fancy and are simple for the child to understand). Show the child the side with the flower and ask them what they see. (They will hopefully say they see the flower). Flip the paper so the flower is facing you and ask the child what they see. (They will hopefully say they see the smiley face). Then, ask them what they think you see. (If they say you also see the smiley face, they have failed).
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Animism (belief that everything that exists has some kind of consciousness): Get a soft toy and hit it multiple times. Ask the child if they think the toy has been hurt, and ask them why. (If they say the toy has not been hurt because it isn't real, or something along those lines, they have passed!)
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Centration (only able to focus on one quality or feature at a time): Arrange tokens/coins into two equal lines of six with equal spacing. Ask them which row has the most tokens. (They will hopefully say both are equal). Now, space the tokens out in the first row and bunch up the tokens in the second row. Then, ask them which row has the most tokens. (If they say the first row has more tokens, they have failed. You will be surprised how many young children fail this test!)
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Reversibility (ability to mentally follow a sequence of events or line of reasoning back to the starting point): Take a deflated ball and pump it. If they can understand that it is the same ball that has been pumped up and now looks different, they have passed.
Concrete operational-
Mental operation (ability to imagine the consequences of something happening without needing it to happen): Ask the child what will occur to an object left outside when it rains. If they can successfully say it will get wet, they have passed. OR ask them to add two numbers without any working out paper or without using their fingers.
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Conservation (certain properties of an object can remain the same even when its appearance changes): An easy test for this is to get 2 glasses that have the exact same shape. Pour water into both of them (colour the water using food colouring to make this easier) so that the amount of water looks visually equal. Ask them which one has more water. (They will hopefully say both glasses have the same volume of water). Then, take a glass that is extremely tall or extremely wide. Pour the water from one of the original glasses into the new glass where the child can see you doing this. Ask them which glass has more water. If they say the glass which is wider/taller has more/less water, then they have failed. If they say they still have the same amount of water, they have passed.
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Classification (ability to organise objects or events into categories based on common features that set them apart from other categories): Get 20 beads (or anything else that can be substituted for beads). Make 18 of them one colour, and the other 2 a distinctively different colour. Ask them how they would group the beads. If they put the 18 beads together and the other 2 beads separately, they have passed. This involves grouping objects based on colour.
Formal operational-
Abstract thinking (thinking that does not rely on being able to see, visualise, experience or manipulate in order to understand something): Write down a simple algebraic equation for the child to complete, such as: 5 + x = 7. What number does x represent? If they can answer correctly, they have passed.
I did this experiment on my younger sibling (thankfully she passed everything with ease!) It'd be interesting to hear others' results!
If you'd like to see this experiment (or parts of it) performed on children, or just want to be entertained, I highly recommend you watch this video. It's full of young children failing at these exercises - not just entertaining though, but also interesting to see the psychological thinking behind their reasoning.