First Response to Engaging Ideas

I approach the Bean book as a teacher of developmental mathematics and the two part course Fundamentals of Mathematics I and II for Elementary Teachers.   

I enjoyed reading Beans’s description of the writing process as a tool for thinking.   He writes that “writers produce multiple drafts because the act of writing is itself an act of problem solving.”   This has definitely been my experience with writing, and it’s nice to hear someone articulate it so well.    It’s taken me some time to write this, but I feel that I have begun thinking through some issues involved in applying writing to mathematics classes. 

Mathematicians create and write proofs.  This is much like the writing process described by Bean.   Prior to writing a proof, the mathematician must come up with a question, explore and come up with a proposed answer to the question.  They need to have an idea on how to logically show that this is true.  In the process of writing down this idea, they may find that the logic does not hold.  They may find that there is a more clever way to obtain the proof or just a more concise way to get through parts of the logic.  It is even possible to end up disproving your initial hunch.   Mathematics differ from other disciplines in that no one attempts to disprove what already has been proved.   This is because mathematics does not try to understand the world, but merely tries to expand upon something human made.    

The classes that I teach do not ask students to write proofs.  They do ask students to communicate their reasoning.  This is nearly always done through a series of steps using mathematical symbols.  Unfortunately students tend to memorize procedures rather than trying to make sense of them.  Students generally do not appreciate alternative procedures, nor do they attempt to devise their own.  This not only turns mathematics into a excruciatingly dull subject; it also impedes their ability to progress in mathematics.

I believe that writing can help students make sense of mathematics. The text that we use for our teacher classes quotes Einstein as saying  “a description in plain language is a criterion of the degree of understanding that has been reached.”  I believe that this is the case, but I am not sure how to get my students to buy into it.    

Two frustrations in assigning writing:

  

   1) Valuing writing in math class

Students simply see any writing in mathematics class as a needless and annoying diversion from the bottom line of applying a procedure to obtain the correct answer.  Something as simple as using a complete sentence for the answer to a word problem is frequently ignored.   Without the sentence, it is difficult to determine whether the student understood the question.  The idea of asking my developmental students to write a complete explanation of how they solved a problem, using Polya steps for example, seems daunting.

My students who plan to be elementary school teachers are more open to writing.  They realize that they need to work on communication.   Unfortunately, what they feel they need to communicate is a simple way for children to perform procedures.  

   2) Challenging students to get beyond procedures

As much as teachers try to emphasize that mathematics should “make sense” and the student’s job is to play around and struggle with it until it makes sense to them, most students resist this.  A major theme of our mathematics for teachers course is the difference between the “how” and the “why”.  Nevertheless, at final exam time many students persist in stating the steps of an algorithm rather than providing meaning to those steps.  In a sense they are giving me the “And Then”  when I have asked for some critical thought.   

The assignments and tests that we typically employ in developmental mathematics classes often allow students to get by largely through memorizing procedures.  I plan to focus on one developmental course this summer.  I think that I can come up with assignments that fit most of Bean’s criteria.  What I hope to get from the book are ways to communicate to students what I am expecting of them and a means to provide constructive feedback.   Although I am convinced that some of students’ difficulty comes from a minimalist approach to studying, I think Bean is correct that other factors also come into play.  I would like to try to identify what these factors are in mathematics writing.   I hope to also apply this to the writing assignments that I already use in my teacher courses.