Thursday, March 03, 2005

The $64,000 Question

Resident edusphere psychometrician Kimberly Swygert asks the $64,000 question:

The notion that, somehow, children can be very educated and advanced and free-thinking and logical, yet unable to handle a test of basic reading, science, and math skills, is so pervasive these days. Where did this meme come from? When did we decide that it was more important for fifth-graders to have these "critical thinking" skills than to understand how many days there are in a year, or be able to summarize the main point of a three-page story?
I don’t know where this came from, but I’ve said before that I don’t believe it. (Click here for more on that.) Now, I’d like to open up a new line of attack on this notion, this time using the tenets of logic and reason to disprove it.

First, let me begin with a math problem taken from Texas’ TAKS exam, relayed by Claire in Kimberly’s comments section:
“An ice cream shop sold 600 scoops of ice cream in a day. They used half of the scoops to make banana splits. The other half was used to make single and double dip cones. Approximately how many cones did they sell? Possible answers were: a) 300 b) 200 c) 100 d) 50”

The 'correct' answer was supposed to be a) 300. My daughter put in b) 200, and got it wrong.

This would be correct if they sold 300 single-dip cones. But suppose they sold 150 double-dip cones? In that case, the correct answer would be b) 200, as that is closest.

The error is in the assumption that # of scoops = # of cones, which is not necessarily true.
As Claire points out in her comment, this problem reinforces very sloppy reasoning. How? It forces students to proceed from unclear premises. It neither defines what single-dip and double-dip cones are nor says if the shop sells more single-dip or double-dip cones. If a singe-dip scoop takes one scoop and a double-dip scoop takes two scoops, then 300 is a less reasonable answer than 200, since it assumes (as Claire states) that the store sold all of one kind of cone. The problem with the above problem is that we just don’t know. My response to the above would be that there is not enough information to solve the problem.

If I were to ask you to think critically to solve the above problem, upon what skills would you draw? I would say that mathematical knowledge would be on the list, as would reasoning skills. The problem with reasoning skills is that a true outcome depends upon true and clear premises. How would one determine that the premises in a math problem are clear or unclear? He would use mathematical knowledge. He would also have to know that he had to make sure the premises were clear.

So let us do a little reasoning of our own. Assume that one must start from true premises when reasoning and assume that it takes knowledge of the subject matter to determine whether a premise is true. Does it not follow that it would be impossible to reason without knowledge of the subject matter?

Let us explore this a little further. I, for example, have very little knowledge of psychometrics aside from what I’ve read on Kimberly’s blog. Therefore, if Kimberly were to lay out a piece of reasoning about psychometrics, I would have no way of verifying whether it lead to a true conclusion or not, since I have no way of evaluating the premises. While I could look for logical fallacies within her argument, it would not help me since they can only prove that the argument is faulty, not that the conclusion is true or false. In fact, a fallacious argument can reach a true conclusion.

Now, let me lay out a piece of reasoning about common practice harmony. Assume that parallel sixths are forbidden by the rules of common practice harmony. Assume that any composer from 1600 to 1900 who broke the rules were bad composers. Bach, Beethoven, Mozart, and Haydn all lived between 1600 and 1900 and used parallel sixths regularly, therefore they are bad composers.

I’d reckon that most of my readers haven’t taken music theory and therefore have no way of judging whether my premises are true. My conclusion seems to be ludicrous, since the four composers named are the four most famed composers from that period. My conclusion is logically solid, though it starts from a false premise. Parallel sixths are not forbidden. In fact, they are encouraged. Therefore, a false premise produced a false conclusion.

In summary, being able to judge the truth of premises is vital to reasoning. To me, critical thinking and reasoning go hand in hand, therefore the above is relevant. This does not hold true for everyone, though, and that is the problem.

Now, to address Kimberly’s point, we need a definition of critical thinking. I’ve never heard anyone who’s employed the term “critical thinking” clearly define what it is. The best definition I’ve found is that presented by the Critical Thinking Community, though it is still not clear to me since it employs a lot of terms which have not been defined. Here is their definition, which comes about 3/4 of the way down the page:
A Definition

Critical thinking is that mode of thinking - about any subject, content, or
problem - in which the thinker improves the quality of his or her thinking
by skillfully taking charge of the structures inherent in thinking and
imposing intellectual standards upon them.
What are the structures inherent in thinking? What constitutes quality thinking? What intellectual standards should be imposed? (NOTE: I could not a satisfactory answers anywhere on the page that contained this definition.)

In this case, the best answer I can give to Kimberly’s point is that reasoning is dependent on basic skills while "critical thinking" is too poorly defined to work with as a concept.

As for the question of where it came from, I wonder what the correlation is between those who hold the notion Kimberly describes and those who believe that truth is relative. That just might be where the answer is... (Read back soon for my thoughts on this.)

TOPIC: Education