Sunday, January 17, 2016

How to Demonstrate an Understanding of a Concept

Recently, the Chinese economy and its growth have been in the news a lot.  One thing that caught my attention was the statement that the rate of growth in the Chinese economy had slowed more than was expected.  I found that interesting because my mind immediately went from the statement to thinking about third derivatives.  Why, you ask?  In part because even after all these years, I remain a math geek.  More importantly, because the way I teach, I am always interested in ways to figure out whether people understand the concepts rather than just are capable of remembering a formula.  

If I wanted to test students on third derivatives, I could, of course, give an equation and ask the students to calculate.  But what if I were to ask the following question:
The news reported that investors were spooked about the Chinese economy.  While the economy is still growing, the rate of growth slowed more than expected form the middle of 2014 through the end of 2015.  Draw a graph that is consistent with this comparison of two growth paths.  From the description of the news, what, if anything, can you say about the signs of the first, second, third, and fourth derivatives.
This would be a cool way to see if people can visualize something and relate it to a mathematical concept.  It shows the ability to translate the concept into something meaningful to help with an interpretation.  By asking about the fourth derivative, a student would have to demonstrate an understanding of where the information ends.

I always like to test students on their capacity to apply ideas to real like.  I am much more interested in that type of understanding and learning than in whether they can plow through equations.  Of course, the highest level of expertise means doing both.  But I think that sometimes we train people to do far too much on equations and not enough on understanding what it all means.

And I am someone who cares about what it all means.  If it means nothing, then why bother learning it in the first place?