Thoughts on Critical Thinking in Mathematics

This is my response to this comment by a reader of The New York Times.

First of all, as a math instructor, I’m a little offended that this reader has called our education system pathetic without actually giving a solution to the problem.  Although, I understand where the reader is coming from in terms of needing a solid foundation in order to do higher mathematics, I want to share a share a story with you based on what I have learned from working with students whose first experience with mathematics was outside of the United States.
Last semester I had a student in my basic mathematics class who tried as hard as she could in the class.  She could do most of the arithmetic mentally, or by hand.  She was especially good at calculating fractions mentally because she followed a memorized formula for doing so.  She was used to being taught the formulas upfront and then shown some ways of how to apply the formulas.  I always discourage students from memorizing formulas for calculating fractions because it’s exactly that.  It’s nothing more than memorization that absolutely does nothing to help the person understand what a fraction actually is.
Throughout the entire semester, she was at the top of the class.  However, when the test came along, I had a question that made the students think about what a fraction really is. A question in which one actually had to know that a fraction of something, whatever it is, is a fraction of something that has been split up into a certain number of equal parts.  Somewhere in the process of the rote memorization that this student had done, she missed the point of equal parts, and tried several times to break an object into several unequal parts.  No one else missed this problem.  Because they took their learning experience beyond just simply memorizing a formula for calculating fractions that didn’t apply to every problem that involves fractions.
And that is where I think that this reader is missing the point.  I think that by having students think critically and develop the formulas for themselves, that certain learning occurs and details will be learned through the discovery of these formulas that might be missed otherwise.  I like to think of formulas as shortcuts for people who are too lazy to derive something for themselves.  Of course formulas are nice, but it’s always nice to know where the formula is coming from.  How can someone honestly say that they know where the formula is coming from if the only thing they know to be true about it is that someone told them that it holds true and therefore they should memorize it?

Discovery learning takes a long time, and you won’t get the answer right away.  But getting the answer right away is not the point of the whole thing.  And just because you don’t get the answer right away, doesn’t mean that math is difficult and impossible to understand.  It doesn’t mean that you should give up and blame the entire educational system.  You need to understand that true learning and true understanding takes longer to foster because the roots of it go deeper than any foundation that is set down using any other method of learning.

Remember the example of my student?  Her so-called solid foundation in the formulas that she memorized about fractions at a young age, her rock solid foundation, was crumbled in just seconds by one question asking her to actually interpret the meaning of what a fraction meant.  To me, this is sad, and I would never want this to happen to anyone that I know.