# Doing mathematics-the difference between deep and superficial knowledge

Math education does not need reform as it equates to rearranging chairs on a sinking Titanic. We need to build a new ship. (Source)

If I am advocating anything, it is only the obvious (and time-tested) idea of “learning by doing.” If I have a method, it is only to convey my love for my subject honestly, and to help inspire my students to engage in a delightful and fascinating ADVENTURE – to actually do mathematics, and to thereby gain an appreciation for the depth, subtlety, and yes, utility, of this quintessentially human activity. Is that really such a strange and radical idea? Have we really reached a point where one has to argue for teaching that “awakens and stimulates students’ natural curiosity?” As opposed to what? I thought that was the definition of teaching!”

Paul Lockhart– (Source)

Having gone through the process of “doing mathematics” as an undergraduate math major at a nondescript university, I am fully aware of math’s inherent difficulty. Thank God for professors who graded on a curve! This was their way to compensate for a student not deeply understanding the math they were offered. I still get a cold chill when I think of my experience with a course in Fourier analysis that I suffered through in college despite the fact I got an A in it. Howard Gardner called this phenomenon the “correct answer compromise.” (See references.) The teacher/professor assumes that achieving mastery of a topic is beyond what students can do, so the instructor offers a compromise that the students can achieve. (In college grade curving helps the professor who wants to test for mastery avoid having to fail the entire class.)

I was reminded of this when I was writing a lesson (adventure) on using Buffon’s Needle experiment. (Read the lesson to better understand what I write about next.) Usually this activity is done with high school students because to understand it properly you need some experience with calculus (integration) and converting from angles to radian measure. Now both of these powerful ideas can be taught straight forward in lecture format and students can easily get an A without really understanding the concepts in the context of this experiment. In writing the lesson I wanted to avoid using calculus to determine the area under the curve y=.5sin(ø). Instead I programmed (in Scratch) a dart throwing simulation that results in showing me that the area was about 60% of the rectangular dartboard which is the approximate probability of a toothpick crossing one of the lines in a set of parallel lines. It was in the process of writing the program that I began to understand the experiment and its conclusions more deeply.

Once again Paul Lockhart:

“If I have a method, it is only to convey my love for my subject honestly, and to help inspire my students to engage in a delightful and fascinating ADVENTURE – to actually do mathematics, and to thereby gain an appreciation for the depth, subtlety, and yes, utility, of this quintessentially human activity.”

**References**

“Most schools have fallen into a pattern of giving kids exercises and drills that result in their getting answers on tests that look like understanding. It’s what I call the “correct answer compromise”: students read a text, they take a test, and everybody agrees that if they say a certain thing it’ll be counted as understanding.” (Source)

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