They Never Cease to Surprise Me
Today I powered through a nearly two-inch-tall stack of homeworks and quizzes from the gen-ed class. I'd been putting it off because it's so depressing. The class average has been hovering in the D-range, and they're pretty bad at math. Recently we've been covering the hardest material in the course. We've beed talking about the difference between rational numbers and irrational numbers, proving that the square root of two is irrational, and proving that there are more irrational numbers than there are rational numbers. (Really we prove that there are more real numbers than natural numbers -- same difference, really.) I spent seven lectures on this material; they took the exam on Friday.
After grading the giant stack of homework, I set to grading the exam.
The class average was an A!
Students who three weeks ago couldn't tell a prime from a composite, seem to be able to tell a rational number from an irrational number and to tell a countable set from an uncountable one.
I'm particularly impressed that they can tell rationals from irrationals. This you might not think is a big deal. We introduce rational numbers as any number that can be written as a fraction where both the numerator and the denominator are whole numbers (positive or negative is ok -- just as long as you don't divide by zero); we define irrational numbers as those that it is impossible to write like this. And at first, they're OK with this dichotomy: fraction, yes or no.
But then we give them more information. We tell them about decimals. We tell them that rational numbers have decimal expansions that either terminate or become periodic; irrational numbers have decimal expansions that neither terminate nor become periodic. And here, all common sense goes out the window. They put the idea of "fraction" out of their minds; they reach for their calculators. For most freshmen, a number exists in only the precision that displays on the screen of their calculators. Like a flat earth or a bounded universe, there is no number beyond the edge of the calculator screen. And yet, paradoxically, it also goes on forever, as if carried by some sort of mathematical aether.
This is the point at which they start to tell me that 1/7 is irrational because 1/7 = 0.1428571, which clearly goes on forever without repeating. Similarly, 5/19 = 0.2631579 on the calculator, so it is also irrational in the mind of the freshman. And 3.14 is irrational, too, because to them it equals π, which they have been told is irrational. (As few of them have taken algebra 2, they are unaware of 2.71828 and have no preconceived notions about it.) Every semester I warn them about these numbers and about trusting their calculators; I remind them that the fraction definition trumps everything else. Every semester I trap them with these numbers.
On the exam I asked them whether the square root of 2 was equal to 665857/470832 (and why or why not). Nearly everyone in the class said that they were not equal because the square root of 2 is irrational and is, therefore, impossible to write as a fraction.
After grading the giant stack of homework, I set to grading the exam.
The class average was an A!
Students who three weeks ago couldn't tell a prime from a composite, seem to be able to tell a rational number from an irrational number and to tell a countable set from an uncountable one.
I'm particularly impressed that they can tell rationals from irrationals. This you might not think is a big deal. We introduce rational numbers as any number that can be written as a fraction where both the numerator and the denominator are whole numbers (positive or negative is ok -- just as long as you don't divide by zero); we define irrational numbers as those that it is impossible to write like this. And at first, they're OK with this dichotomy: fraction, yes or no.
But then we give them more information. We tell them about decimals. We tell them that rational numbers have decimal expansions that either terminate or become periodic; irrational numbers have decimal expansions that neither terminate nor become periodic. And here, all common sense goes out the window. They put the idea of "fraction" out of their minds; they reach for their calculators. For most freshmen, a number exists in only the precision that displays on the screen of their calculators. Like a flat earth or a bounded universe, there is no number beyond the edge of the calculator screen. And yet, paradoxically, it also goes on forever, as if carried by some sort of mathematical aether.
This is the point at which they start to tell me that 1/7 is irrational because 1/7 = 0.1428571, which clearly goes on forever without repeating. Similarly, 5/19 = 0.2631579 on the calculator, so it is also irrational in the mind of the freshman. And 3.14 is irrational, too, because to them it equals π, which they have been told is irrational. (As few of them have taken algebra 2, they are unaware of 2.71828 and have no preconceived notions about it.) Every semester I warn them about these numbers and about trusting their calculators; I remind them that the fraction definition trumps everything else. Every semester I trap them with these numbers.
On the exam I asked them whether the square root of 2 was equal to 665857/470832 (and why or why not). Nearly everyone in the class said that they were not equal because the square root of 2 is irrational and is, therefore, impossible to write as a fraction.
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