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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.
11 comments:
"The problem isn't that arithmetic is taught like arithmetic. The problem is that math is taught like arithmetic."
---Anonymous
You can also get that problem from the first chapter of "Math Circles" by Fomin.
Incidentally, if the glasses each consist of only one teaspoon of liquid the outcome is quite different.
At any rate, by "resorting to math", I think you mean doing some sort of calculation. I do think it is true that you aren't really supposed to calculate your way to an answer here. However, I fail to see what this conservation of number principle has to do with it, either. First of all, the way to think about it is as an amount as opposed to a number like the number of molecules, though that would work, too -- it's just an extra complication to the reasoning. And, in either case, the key to understanding this problem is to realize that you will always be taking some of the red substance and putting it back into the beaker it came from when you add the second teaspoon (or to realize something along those lines, at any rate).
There aren't special kinds of knowledge. Knowledge is simply "justified, true belief". (Yes, I've heard of Gettier, just in case there is a wise guy out there.) There isn't flexible or inflexible, domain vs something else. Such categories might make useful psychological models in a certain context, but they aren't real -- they are just our own constructions, created to understand something -- a way of looking at something. This problem is just a logic puzzle designed to get someone to think. The best solution to it doesn't use some particular method or inuitive awareness but rather just ordinary reasoning that is neither flexible nor inflexible and nothing like a calculation, though, either.
And, by the way, that reasoning IS the math. The arithmetic one might do, even if it involves letters and even if those letters are Greek letters, is not math.
Incidentally, if the glasses each consist of only one teaspoon of liquid the outcome is quite different.
No, the answer is the same. The precentage change is equal. The percentage change is always equal; that's where the conservation of number principle comes in. The amount of foreign liquid in each container is irrelevant.
You don't need to know any math to solve the problem if you know the conservation of number principle.
You can even solve a similar problem in which the math becomes exceedingly cumbersome:
I start out with a container with of 1 liter of oil and another container with 1 liter of vinegar. beside me is a box full of different quantity measuring spoons. I randomly select ten spoons and without looking proceed to use each spoon in turn and take a quantity of liquid from container one and put it into container two. I purposely do not mix the liquids well so the mixture is not homogeneous. Then I take the next spoon and take some of the heterogeneous liquid from two and put it into container one. I repeat this procedure until I run out of spoons. At the end I pour an amount of liquid from the more full container into the less full container so that I have equal amounts of liquid in each container. Is the percentage change of foreign liquid in each container the same?
And, by the way, that reasoning IS the math. The arithmetic one might do, even if it involves letters and even if those letters are Greek letters, is not math.
The reasoning is not necessarily the math, although you may be able to reason out the problem mathematically if you do not know the conservation of number principle. If I had given you the modified problem with the difficult math first and you did not know the conservation of number principle beforehand, you'd have a much more difficult time determining the principle using math and your answer would be more prone to error.
Now that I reread the question, it all makes sense. On my first reading, I thought the question was whether or not you could dilute the liquids this way so that there is less red liquid than there is clear liquid in the red beaker (and conversely in the clear beaker). (In other words, there is always more red liquid in the red beaker.)
That's what I suspected.
The ability to apply fundamental laws or governing equations (like Bernoulli's equation, conservation of energy, Maxwell's equations, etc.)to all sorts of problems is very good. In some cases, different fundamental laws or equations can be used to solve the same problem.
Higher order thinking is too vague to be meaningful. In most cases, it refers to some magical process that doesn't rely on any fundamental law or eqation. Students have to just figure it out. It's as if educators believe that there is some sort of magical fundamental law of the brain that has to be developed rather than being directly taught something like the laws of motion.
I thought I knew the answer after reading the question, but all of the explanations have confused me. Here’s my line of reasoning: 1 tbsp of liquid is moved each way, so in the end both have the same total amount of liquid, but you’re moving 100% dye over to the water, and diluted water over to the dye, so the water must have more dye than the dye has water. Is that right?
I’m not sure I get why the discreteness matters. Even if the containers were full of some continuous, real, distribution, instead of a discrete, integral, distribution, wouldn’t “stuff” still be conserved?
d
But, remember, d -- you are moving some of the original dye back over too, so in the end you have not netted a full spoonful of dye in the clear glass. (HAH! I told you all it was all about observing this fact! ;o) )
As it tuns out, each glass actually ends up with the same amount of the other liquid in it.
I'm still confused; I'm going to have to read the original problem again carefully. (I think I'm confused even about what is being asked....)
Higher order thinking is too vague to be meaningful. In most cases, it refers to some magical process that doesn't rely on any fundamental law or eqation.
One of the interesting things about "higher order thinking," to me, is that experts perceive problems in different terms than non-experts do.
I think you'd say that their "schema" is different: they organize the "component parts" of their knowledge (facts, skills) differently.
argh! I should have just taken 10 seconds to write out the math. :-( This is exactly the sort of "shortcut" that would always mess me up in school. Thanks for the explanation.
d
This is exactly the sort of "shortcut" that would always mess me up in school.
I know exactly what you mean -- I basically got it wrong, too, for lack of care on my part. Sometimes, especially in the midst of some big math-is-the-business-of-proving-theorems-not-doing-calculations rant I have this secret anxiety that math really IS just a big game of gotcha after all.
;o)
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