One of Jimmy's favorite toys as a toddler was a simple little bucket of blocks. There were three shapes: a rectangular prism, a triangular prism, and a cylinder. The bucket's lid had three holes: a square, a triangle, and a circle (The picture at right was the only one I could find online -- this sort of toy has gotten much fancier in recent years).
For an adult, it's a simple matter to properly sort the shapes by placing them into the corresponding holes, but for a toddler, it's a real challenge. It took months before Jim was able to put any of the blocks through the holes, despite countless demonstrations by his parents. Maybe parents have a special sort of magic that kids just can't do.
But eventually -- perhaps by accident -- Jimmy managed to get a cylinder through the round hole. Eureka! The puzzle was solved. He put another cylinder through the hole, and another. Aha! All you need to do is put the objects through the round hole. The others are decoys. Next he attempted to put a rectangular prism through the round hole. To his amazement, this didn't work, even when he applied so much force that his little body shook from the effort. How about the triangular prism? This didn't work either. He confirmed that indeed, the remainder of the cylinders fit through the hole. Apparently triangular and rectangular pieces required a different strategy. A smile of realization crossed his face.
Confidently, Jimmy tried his new solution. He removed the lid to the bucket and placed all the triangular and rectangular blocks into the bucket, one by one. Then he replaced the lid and looked to his parents for approval. He had finally solved the puzzle, hadn't he?
It would be several weeks before Jimmy was able to properly sort all the blocks. Even after he had solved the puzzle once and for all, he still played with it for some time afterwards, as if to confirm that the rules of geometry he had carefully sorted out, hadn't changed.
Perhaps surprisingly, it wasn't until 2007 that a systematic study of this simple problem was undertaken. Helena Ãrnkloo and Claes von Hofsten presented 69 infants aged 14 to 26 months with a similar problem while carefully monitoring their attempts to solve it. Instead of just three shapes, they used seven, each progressively more difficult:
Their box was designed to accommodate just one shape at a time. The toddlers were given one block, along with the box and a lid with a hole shaped to match the block, and encouraged to insert it. First the block was oriented vertically, so all the child had to do was pick it up without re-orienting it and place it in the hole. Here are the results:
The 14-month-olds were only able to handle the cylinders with any regularity, and things didn't improve much for the 18-month-olds. But somewhere between 18 and 22 months, nearly all the children showed a tremendous improvement. by 26 months, almost all the shapes were solved quite easily. But this was a relatively simple problem. Next the blocks were lain on their sides, so that some manipulation would be necessary in order to fit them in the box.
Now 14-month-olds did even worse, and there wasn't much improvement at all by 18 months. Again there was a large leap in accuracy at 22 months, except for the triangle block with unequal sides. Even 26-month-olds weren't able to fit this one through the hole. The unequal sides made the block especially challenging, because this meant that it would only go through the hole in one direction. Most kids simply didn't get this concept, and just gave up after trying several times putting the first end they tried through the hole.
Ãrnkloo and von Hofsten took an even closer look at how the toddlers manipulated the blocks, and noticed some very interesting patterns:
- Kids rarely solved the problem by putting the block into the hole and moving it around. Instead they manipulated it in their hands prior to reaching the hole.
- 18-month-olds pre-manipulated some of the blocks and not others, as if in some cases they recognized that a manipulation was necessary and in others they just didn't get it.
- If a block was pre-adjusted improperly, children of all ages rarely recovered from the error. They seemed to "get it" right away or not.
- Older children were more persistent, but persistence rarely paid off
Ãrnkloo and von Hofsten say that this all suggests that toddlers are solving the problem through mental rotation, not by physically manipulating the objects. Interestingly, they found no gender difference in the results. So though there is some evidence that older boys may be better mental-rotators than girls, it hasn't emerged at this age. So this study finds no evidence that any difference in mental rotation ability between boys and girls is due to a genetic or hormonal difference.
Helena Ãrnkloo, Claes von Hofsten (2007). Fitting objects into holes: On the development of spatial cognition skills. Developmental Psychology, 43 (2), 404-416 DOI: 10.1037/0012-1618.104.22.1684
I wonder if kids could be taught to use the physical rotation method rather than a mental rotation method. If the child sees an adult simply pick up the piece and place it correctly, they would naturally assume they would be able to do the same thing. If however, they saw an adult pick up a piece, place it on the hole, and then rotate it, I bet they would be able to learn the 'trick' fairly quickly. This 'trick' should work equally well for all shapes as well. The only possible downside is that without having to rely on mental rotation, would physical rotators have less ability to mentally rotate an object than mental rotators after 26 months...
With n=1, I don't think your solution would work. I definitely tried teaching this to my child by taking the piece she was holding and rotating the piece in place over the correct whole. Eventually she figured out the task on her own roughly on the order of the plots from this paper.
It's also been amazing watching my now slightly older child learn jigsaw puzzle. She's mostly trial and error and very slow, but, one she memorizes the picture, she's amazingly fast.
Interesting. In the past week or so I've been noticing exactly what O & vH describe as I've played similar games with my 15 mo. daughter. She always gets the cylinder right (regardless of how it sits on the floor before picking it up). The other shapes are hit-or-miss. She is pretty good with the square, hexagon, stars, and other shapes that have multiple correct solutions (if each angle is considered a correct solution). I was surprised when I noticed that, if she got the angle wrong, she rarely would rotate the piece to fix it. I guess that shows my assumptions. :)
Similarly, with a large wood-block "jigsaw" puzzle (place the animal blocks in the cut-out holes) she knows which pieces go in which holes, but rarely is able to line them up. The animal shape is painted in the space where the block goes, so she knows that the ladybug goes where the ladybug is painted, but she doesn't seem to understand that the ladybug's head needs to line up in the same direction.
I will have to try Kevin's idea and see if she responds to it. His point makes me realize that we probably don't model failure and how to deal with failure as a part of our play activity. She certainly sees it as she watches us in daily life, but it probably would be good to introduce it so she sees us fail and then try again at tasks that she is learning.
Frankly, I am a firm believer that the reason I excel at packing boxes, Uhauls, etc has less to do with studying solid state physics and understanding rotational groups and more to do with hours and hours of Tetris as an undergrad. So if she has less ability to mentally rotate an object at 26 months, I can introduce her to that addictive game...
i am my family's designated u-haul packer and its greatest tetris master. the next time i see my stepson, he's getting introduced.
Ãrnkloo and von Hofsten say that this all suggests that toddlers are solving the problem through mental rotation, not by physically manipulating the objects.
If we assume Piaget's sensorimotor substages to be true, it would probably be more accurate to say that during the earlier end of the age range examined (at 14 months when tertiary circular reactions take place and infants actively explore the properties of objects), infants primarily solve the problem by physically manipulating objects. But as they get older, they resort to a mental strategy (i.e., after developing object permanence and begin to have dynamic representations of objects in the mind).
The example with Jimmy seems to suggest a Piagetian sensorimotor substage pattern starting with trial and error physical manipulation of objects (secondary circular reactions; usually happens from 8-12 months), and the sudden solutions probably are due to the development of object permanence and formation of dynamic mental representations (usually 18+ months).
Speaking of mental rotation, is it my imagination, or is it the case that for the first toy pictured, the triangular hole is unnecessary? That is, the triangular prisms have a square footprint and will fit nicely through the square hole?
No, that doesn't work. The square on the side of the triangular prism is much larger than the square hole.
#6 I'm sure the toymakers have thought of that and made it the triangle have a base that's slightly broader than the diagonal of the square.
I love this stuff. There's some great work done on naive physics, or how kids think the world works before they are taught or shown differently. My favourite example was an experiment where kids had to report which of two similar objects dropped from the same height hit the ground first.
The only difference between the objects is that one was heavier than the other. As we (should!) know as adults, as Galileo theorised centuries ago and as has been tested on the Moon, the mass of an object doesn't affect how fast it falls when air resistance is taken out of the equation.
The manipulation in this experiment was brilliant. In the first condition, the experimenter picked up the objects and dropped them - kids were at chance. But when the experimenter asked the kids to pick up the objects, they consistently said that the heavier object hit the ground first, even when it clearly didn't.
This research brought to you by the University of Bristol.