Piaget's theory states that a child on the preoperational stage can only think in one dimension. This would be that when thinking about size, they only think of one dimension. Or in terms of a question, they can only focus on one variable.

I don't think this is true, I think they can handle 2 (at least)

OK, lets think about the beaker problem. The kid is shown 2 identical beakers filled with the same quantity of a liquid. Then all the liquid from one beaker is poured into a second but thinner and taller beaker. The kid is asked which beaker has more liquid in it. The kid ALWAYS says the taller one (even though they have the same amount.)

beaker A	beaker B
30 x 20 (circumference*height)	15 x 80 (circumference*height)

Why is this??? Piaget would say that the child can only focus on one dimension of the beaker, and since the beaker got taller, the child only focused on this aspect and said there was more liquid in the taller beaker. My question is why the kids ALWAYS say the taller one has more liquid. If they can only focus on one dimension, why don't they ever say "the wider one," and ignore that the other one is taller????

So What's my theory?

I believe kids think in 2 dimensions (area). With a 3 dimansional object, like a beaker, they will see the lateral area, meaning how much space the surface of the object occupies. In the case of the beaker problem, the lateral area of beaker b is a bit higher. (Lateral area of beaker a = 30 x 20 = 600; Lateral area of B = 15 x 80 = 1200). This makes beaker BLOOK bigger, because the object is bigger, but it uses space less efficiently so it holds the same amount of volume as object A. (Object B uses more glass to hold the same amount of volume). According to a child at the preoperational state, the taller beaker looks bigger, so it must have more liquid. This would explain the consistency of the answers.

Actually, I think EVERYBODY'S first impression of beaker B is that it has more liquid (if we never saw anybody pouring the liquid from one beaker to the other). That is, until we put the knowledge learned in high school geometry class to use.

Let us see this idea portrayed in a different problem...sheets of paper

Quick question, which row has MORE squares??

A.

B.

Did you answer "B"? I sure did. That is, before I thought about it. Most adults would say they have the same amount because they automatically assume it is a trick question...but come on, truthfully, WHICH ONE LOOKS LIKE IT HAS MORE SQUARES? According to Piaget, only kids at the preoperational stage say it is "B" because they only pay attention to one dimension. The truth is that row "B" takes up more space....That's 2 DIMENSIONS, not one. I had to count the squares to double check I which one had more squares.

When you were a child, do you remember any occations where you, or a budy of yours, had a bunch of coin money and spread all the money on the ground to make it look like you had more money? If you have a nice pile of quarters in a single column and your buddy has the same pile spread out on the table, your buddy will seem to have more quarters because the money is taking up more space (more area = less space efficiency; area = 2 dimensions).

I guess you can call this a visual effect.

First: Minds at the preoperational stage can't undo problems in their head, "irreversible thinking." So if we go to the beaker problem... if we pour the liquid from a short beaker to the tall beaker before we ask THE question is pointless....they cant reverse the problem in their heads. All they see are a tall and a short beaker filled with fluid.

Second: Kids always seem to answer the question saying that the taller beaker has more liquid. "Why not the wider one?" Well, it's simple...they are only paying attention to 2 dimensions....base AND height. Unless people have studied about Volume in their High school or Middle school Geometry classes, they don't fully understand the concept of Volume....Base*Width*Height

Kids will look at the beakers and see Base times Height...the 2 dimensional area (almost thinking in Lateral area) If one thinks about it, the taller beaker has more lateral area, it uses up more space. If you drew a picture of both beakers, proportionally correct, when you color in the liquid, you'd use up more of the color pencil on the taller beaker. Even at first glance, if you ask ANYBODY (I think) Which beaker contains more liquid, (with out pouring any liquid from one beaker to the other) Most people would probably say the taller one has more liquid...then they would think about it for a second, remember their past experiences with Geometry or Piaget experiments and realize that maybe they contain the same amount of liquid, and if they think about it TOO much they might even think that the taller one has more liquid.

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