In part one I asked you to use your higher order thinking skills to solve this problem:
You have two identical glasses, both filled to exactly the same level. One contains red dye, the other water. You take exactly one spoonful of red dye and put it in the water glass. Then you take one spoonful of the mixture from the water glass and return it to the red dye glass.
Question: Is there more red dye in the water glass than water in the red dye glass? Or is there more water in the red dye glass than red dye in the water glass? In other words, the percentage of foreign matter in each glass has changed. Has the percentage changed more in one of the glasses, or is the percentage change the same for both glasses?
I even gave you a hint:
Instead of water and red dye, think of red balls and white balls.
Assume that each glass starts out with 100 balls of a single color. Now remove a number of red balls from the red-ball glass and put them in the white-ball glass. Then return the same number of balls from the glass with the “mixture” and put them in the red-ball glass. Do this with different numbers of red and white balls.
Now it's time to end your suffering and give you the answer.
To solve the problem (without resorting to math) you need to understand the concept of conservation of number--a Piagetian concept. According to Piaget, conservation of number is
the understanding that the number of objects remains the same when they are rearranged spatially.
Also according to Piaget, you should have developed this concept naturally by the age of seven:
Piaget proposed that number conservation develops when the child reaches the stage of Concrete Operations at around 7 years of age. Around this time, children also develop an understanding of other forms of conservation (e.g., weight, mass). However, number conservation is often the first form of conservation to develop. Before the stage of Concrete Operations, children may believe that the number of objects can increase or decrease when they are moved around.
In Piaget's classic example he arranged objects in two rows, then spread out the objects, and asked the child if the number had changed. Apparently, children younger than about seven don't realize that the number of objects stays the same, i.e., their number is conserved.
No doubt, as an adult you understand the concept of conservation of number and could answer Piaget's problem with the rows of objects readily. But why couldn't you answer my red dye and water example? It's based on the same concept. Liquids are made up of a fixed number of molecules whose number is conserved. So why the difficulty?
What I was trying to do was demonstrate the difference between flexible and inflexible knowledge. You understand that number is conserved and can solve problems involving concrete objects with ease. However, you may have struggled with the unfamiliar red dye and water problem I presented, not realizing that liquids are composed of molecules. Even those of you that solved the problem by resorting to solving a math ratio problem may not have realized that the problem is readily solvable by knowing the number is conserved without resorting to math.
Hopefully, this demonstrates that most people do not have a flexible understanding of conservation of number that is readily generalizable to unfamiliar examples. Your knowledge of conservation of number is likely not flexible. It did not develop naturally like Piaget said it would and it probably was not taught well to you either. The concept remains inflexible.
Answering the problem doesn't require higher order thinking skills or critical thinking skills or any other fancy jargon educators like to use. All it requires is a flexible understanding and the application of a basic concept that any seven year old readily understands--conservation of number. If the basic concept/skill is well taught to the student and the student is given sufficient practice, the student will eventually develop a flexible understanding of the concept and will be able to apply the concept to solve tricky problems. You don't need a super high IQ to understand such things if you were taught them beforehand.
However, if this concept is not well taught, the student must rely on other knowledge he may have learned to solve the problem indirectly. Setting up a math ratio problem to solve the problem may be one way to solve the problem without having a flexible concept of conservation of number. I bet many of you math heads solved the problem this way, demonstrating your flexible understanding of the concept of ratio--another basic concept.
But what about higher order thinking skills. Do these even exist? Or are they just the byproduct of not receiving good instruction of basic skills?
NB: I got this problem from pp. 7-10 of chapter 4 of Engelmann's book.