Once again, a possible big lesson from the latest calculus homework set. One exercise gave the students this limit:
They were asked to figure this limit three ways: By graphing the function in a small window around x = 0 and estimating the limit; by making a table of values for the function near x = 0; and by doing some fancy algebra tricks to compute the limit exactly.
Most students successfully estimated the limit to be about 0.2887 using the graph and again using the table. (Although a lot of students figured that they didn’t need to include the graph they used, I guess because I should take their word for it?) To repeat: in parts (a) and (b), the students prove fairly conclusively that the limit does in fact exist and is equal to something like 0.2887.
Then they try it the algebra way, and all hell breaks loose. People squared the top and bottom of the fraction, thinking that this is a rule. (Question: Is 2/3 the same as 4/9?) Then they carry out the squaring in the numerator to get 3 + x – 3. Or perhaps they ignore the stern warning I gave them about just taking the limit of top and bottom of a fraction without first checking to see if the denominator vanishes. One way or another, they get that the limit fails to exist. In complete contradiction to their first two calculations.
Critical thinking in this situation would lead a student to have the following thought: Wait a minute. I got the limit to fail to exist. But that can’t possibly be right, because my first two calculations are telling me that there is a limit, and it’s equal to about 0.2887. Clearly something isn’t right with my algebra. I had better go back and fix it.
Uncritical thinking in this situation would give us this: Done with that one. On to the next one.
[Update: Or, suppose you found the instantaneous velocity of an arrow shot upward on the moon at a certain time to be, say, 57,942 meters per second. Yes, that’s a comma and not a decimal point. That speed is equal to 129,613 miles per hour. The speed of sound at sea level is 761 miles per hour. In other words, this arrow is moving a little over 170 times the speed of sound. Something in you should react to this, saying, “WTF have I done?” And then you go back and fix it. ]
I’m thinking more and more that “critical thinking” is nothing more than having an intuition about when something can’t possibly be right, having the good sense to pay attention to that intuition, and then having the combination of skill and desire to get things right which compels the student to go back and get his or her hands dirty. In fact I think only a part of “critical thinking” involves thinking — a good deal of it appears to be a combination of intuition and intellectual courage.