It might be worth attempting to see how you perform on certain types of spatial thinking problems that most people claim to use imagery to solve (although no correlation seems to exist between spatial thinking ability and the vividness that people report their imagery to have). Try to solve the problem below, in your head, without drawing diagrams or making calculations on paper or anything like that.
The four narrow sides of a 1 cm by 4 cm by 4 cm block are painted red. The top and bottom are painted blue. The block is then cut into sixteen 1 cm cubes.
How many cubes have both red and blue faces?
How many cubes have one red and two blue faces?
How many cubes have no painted faces?
Most people say they use imagery to do this, and count the relevant cubes in their image. Were you able to solve all or any part of the problem at all? Did it seem very difficult? How, in fact, did you solve it (if you did)? Did you have to consciously employ any formal knowledge of geometry or other mathematics (beyond counting)?
When I solved this, I had the interesting experience of Imagining the 4x4 block of 16 blocks, noting that the outside ones (all but 4) had red paint on them, and all of them had blue paint… but I only “put” blue paint on the top. My diagram was flat, oriented like a pancake. None of this was Mental Imagery. Then when I was asked how many cubes had red and blue faces, I felt around the edges of the block. Motor/haptic mental imagery. Then when I was asked how many cubes had 1 red and 2 blue faces, I immediately thought the question was 1 red and 1 blue since I didn’t have blue paint on the bottom in my model (I’m not sure if I had a bottom in my model). I thought “when would they have more than 1 red? ah the corners”, and then had the distinct vivid motor mental imagery of moving my hand and touching two non-corner side blocks on the left of my model, then two at the far side, then two on the right, then two on the near side, counting “2, 4, 6, 8”. This was a different experience than my usual Imagining… but I’m not sure if it was qualitatively different or just more “vivid” motor mental imagery.
Did anyone else have trouble recalling the red vs blue sides? (based on my experience with this (below), it seems as though my mental association was essentially “top and bottom are same” and “narrows are same” but neither really had a color. When I close my eyes, I don’t see “red” or “blue”)
At first I was imagining a 1cm by 1cm by 4cm block. I then realized that getting the 16 cubes of 1cm each out of this wasn’t possible and then went to the accurate idea of a 4 by 4 by 1. I realize I am having a great deal of trouble going from a 1cm by 4cm rectangle and then adding depth, while the 4 x 4 square I can add the 1cm of depth much easier. I can rotate the image of the 4x4x1 around in my head and yet cannot do the same with the 4x1x4 (despite the fact that I recognize that they are the same image).
Adding colors: four narrow are red, two fat are blue. Got it. Break it up into cubes.
Red and blue faces requires it touch the outside of the 4x4 square. 4 along top side, four along bottom side, and two each on the left and right (already counted the corners).=12 total red and blues.
One red and two blue… which ones were red and which were blue again? I know that the four narrows are the same and that the top+bottom are different… and just logicked that if the question is one red and two blue, that means the narrows were blue (the reds don’t touch->top and bottom). To be two red and one blue… WAIT—that logicking doesn’t work because the edges have Top side bottom. So I’ll look back at which sides were which color. In my mind I just have “top and bottom are same color” but that’s not assigned to red or to blue. Okay—top and bottom are blue. This means along the edge of the 4x4. uh… the perimeter again. So 4+3+3+2=12. Unless you’re asking for two red and exactly one blue, in which case it’s the corners… wait—two red? red was the narrow side color. i think i switched them again. Final answer: to have two red and one blue you must have two red, meaning the narrow side meets a narrow side, which only happens in four places. Four.
Just re-read the question. You asked for one red and two blue. All of them except the middle 4 blocks, so 12.
No painted faces? All of them are painted on at least one side: we painted three sides of the block, each of the 16 cubes we chopped the block into touched at least one side. 0 are completely uncolored.
This is very interesting. I am having trouble understanding the experience of imagining the 4x4 block of 16 blocks well enough to note that there are four interior blocks w/o red paint on them without picturing them.
I could imagine that this could be done with just logic (reasoning about how many blocks there must be in different categories, which is maybe how I would do it if the problem were more complex, or took place in four dimensions for example), but you said you had a diagram...
So it sounds like you did have mental imagery, it was just 2-d instead of 3-d.
But apparently that wasn’t very vivid, because you still had to do the haptic imagery thing. How vivid is the experience of this motor mental imagery for you? I’m wondering if I’m missing out on that in the way that you’re missing out on more vivid visual mental imagery.
I was given an excellent geometry problem by Dr. Nigel Thomas.
When I solved this, I had the interesting experience of Imagining the 4x4 block of 16 blocks, noting that the outside ones (all but 4) had red paint on them, and all of them had blue paint… but I only “put” blue paint on the top. My diagram was flat, oriented like a pancake. None of this was Mental Imagery. Then when I was asked how many cubes had red and blue faces, I felt around the edges of the block. Motor/haptic mental imagery. Then when I was asked how many cubes had 1 red and 2 blue faces, I immediately thought the question was 1 red and 1 blue since I didn’t have blue paint on the bottom in my model (I’m not sure if I had a bottom in my model). I thought “when would they have more than 1 red? ah the corners”, and then had the distinct vivid motor mental imagery of moving my hand and touching two non-corner side blocks on the left of my model, then two at the far side, then two on the right, then two on the near side, counting “2, 4, 6, 8”. This was a different experience than my usual Imagining… but I’m not sure if it was qualitatively different or just more “vivid” motor mental imagery.
Did anyone else have trouble recalling the red vs blue sides? (based on my experience with this (below), it seems as though my mental association was essentially “top and bottom are same” and “narrows are same” but neither really had a color. When I close my eyes, I don’t see “red” or “blue”)
At first I was imagining a 1cm by 1cm by 4cm block. I then realized that getting the 16 cubes of 1cm each out of this wasn’t possible and then went to the accurate idea of a 4 by 4 by 1. I realize I am having a great deal of trouble going from a 1cm by 4cm rectangle and then adding depth, while the 4 x 4 square I can add the 1cm of depth much easier. I can rotate the image of the 4x4x1 around in my head and yet cannot do the same with the 4x1x4 (despite the fact that I recognize that they are the same image).
Adding colors: four narrow are red, two fat are blue. Got it. Break it up into cubes.
Red and blue faces requires it touch the outside of the 4x4 square. 4 along top side, four along bottom side, and two each on the left and right (already counted the corners).=12 total red and blues.
One red and two blue… which ones were red and which were blue again? I know that the four narrows are the same and that the top+bottom are different… and just logicked that if the question is one red and two blue, that means the narrows were blue (the reds don’t touch->top and bottom). To be two red and one blue… WAIT—that logicking doesn’t work because the edges have Top side bottom. So I’ll look back at which sides were which color. In my mind I just have “top and bottom are same color” but that’s not assigned to red or to blue. Okay—top and bottom are blue. This means along the edge of the 4x4. uh… the perimeter again. So 4+3+3+2=12. Unless you’re asking for two red and exactly one blue, in which case it’s the corners… wait—two red? red was the narrow side color. i think i switched them again. Final answer: to have two red and one blue you must have two red, meaning the narrow side meets a narrow side, which only happens in four places. Four.
Just re-read the question. You asked for one red and two blue. All of them except the middle 4 blocks, so 12.
No painted faces? All of them are painted on at least one side: we painted three sides of the block, each of the 16 cubes we chopped the block into touched at least one side. 0 are completely uncolored.
This is very interesting. I am having trouble understanding the experience of imagining the 4x4 block of 16 blocks well enough to note that there are four interior blocks w/o red paint on them without picturing them.
I could imagine that this could be done with just logic (reasoning about how many blocks there must be in different categories, which is maybe how I would do it if the problem were more complex, or took place in four dimensions for example), but you said you had a diagram...
So it sounds like you did have mental imagery, it was just 2-d instead of 3-d.
But apparently that wasn’t very vivid, because you still had to do the haptic imagery thing. How vivid is the experience of this motor mental imagery for you? I’m wondering if I’m missing out on that in the way that you’re missing out on more vivid visual mental imagery.