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This is the seventh in a series of articles from an ed school student working towards certification as a math teacher.  (Click for his first, second, third, fourth, fifth and sixth missives.)  As always, he prefers to remain anonymous. -ed.

One summer during college I had a brief stint working the night shift at an all night drugstore in a rather scary section of town.  On my first night, while waiting for someone to buy something or shoot me, the dead drunk security guard for the store came over and introduced himself.  He put his arm on my shoulder and muttered something that I couldn’t make out.  I kept asking him to repeat himself which made him angry until he shouted: “I said you don’t have to worry about anything with me around.”  I did not find this clarification reassuring.

This situation reminds me a lot of what I’m going through in ed school.  I am confronted with explanations I can’t quite comprehend, but whose clarifications upset me further.  A case in point is the textbook we are reading in my math teaching methods class.  The textbook is “Teaching Mathematics in Secondary and Middle School” by James S. Cangelosi.  Excerpt from Chapter 4:

“Because mathematics is widely misunderstood to be a linear sequence of skills to be mastered one at a time in a fixed order, some people think teaching mathematics is a matter of following a prescribed curriculum guide or mathematics textbook.  … “

That would be me.  Sorry, but I find that teaching the distance formula before delving into what is the Pythagorean Theorem, omits necessary logic and structure.  Or teaching the quadratic formula first with derivation later, or in some cases, no derivation at all.  That this type of “anything goes” technique with no regard to mastery is embraced by those who decry the practice of giving students formulas to memorize without understanding underlying concepts is also disturbing.  Given how Cangelosi believes mathematics is “widely misunderstood”, however, I would guess that I’m not alone in my beliefs.  He goes on: 

“Textbooks present information and exercises on mathematical topics, but typical textbook presentations are pedagogically unsound from a constructivist perspective. … Thus, textbooks should be used only as references and sources of exercises–not religiously followed page by page.”

I think I’ve talked about constructivism enough for you to know my reaction to that.  He concludes his rant with the following: 

“Word problems from mathematics textbooks provide convenient exercises for students to experience some–but not all–aspects of real-life problem solving.  With a real-life problem, students are confronted with puzzling questions they want to answer.  Textbook word problems…present puzzling questions, but rarely are they questions students feel a need to answer.” 

This brings up the issue of just what a “real life” problem is and why it’s different than the traditional ones the author eschews.  Interesting that he feels students rarely feel a “need” to answer textbook word problems.  I’ve been observing classes at a school in which the math teachers teach religiously from Dolciani’s algebra textbooks (written in the late 60’s and then revised in the 80’s and very effective at teaching algebra to mastery).  The students I observed at the school find the word problems in Dolciani challenging.  In the spirit of full disclosure, these are honors and “gifted and talented” students, some taking algebra in the 7th grade.  I tell you this for those of you who believe that mastery and higher order thinking skills come naturally to bright kids anyway and they feel a “need to answer” everything.   

In addition to the pronouncements made in the textbook, Mr. NCTM handed out a one-page excerpt from a paper at the end of class a few weeks ago and said we would discuss it next session.  It was an essay against the “traditional” word problems in algebra in which the unidentified author stated that such problems “convince students that there are no real applications of algebra, since the problems are so ridiculous.”  He gave an example of a work problem: John shovels snow from a walk in 4 minutes; Mary can do the same walk 3 minutes.  How long will it take them to finish the job together?  The author rails that no one can shovel snow that fast.  Change it to 30 and 40 minutes if it bothers you so much; the concept is still the same.  But the author is not concerned with that.  The author of the essay finds algebra problems to be such that students will ask, “Who cares what the answer is?”

Like hearing what the drunken security guard at the drugstore was trying to tell me, I dreaded what Mr. NCTM would say about the essay.  I fully expected a facilitated class discussion ending with the conclusion that short-term relevance trumps content and mastery—problems that are messy and time-consuming like finding the best long distance telephone plan are much more instructional.  To my surprise, the essay was not discussed.  Mr. NCTM said only that it was written by a “very smart mathematician” at University of Chicago in the 1980’s, a man by the name of Zal Usiskin.  For those who don’t recognize the name, Val Usiskin was a major player in the development of the Everyday Mathematics program which is used in K-6.

He is most likely responsible for the following which appears in the Teacher’s Reference Manual for that program: “The authors of Everyday Math do not believe it is worth the time and effort to develop highly efficient paper-and-pencil algorithms for all possible whole number, fractions and decimal division problems.…It is simply counterproductive to invest hours of precious class time on such algorithms. The math payoff is not worth the cost, particularly because quotients can be found quickly and accurately with a calculator.”

If I may add my own clarification to both Cangelosi (who has a masters in math) and Usiskin: In other words, the U.S. doesn’t need to produce scientists and engineers when we can hire them more cheaply from India and China where traditional word problems are presented…and solved with alacrity.  Whether Messrs. Cangelosi and Usiskin need good swift kicks is something I will let the reader discover in true constructivist spirit.

With no further clarification, I remain faithfully yours,

John Dewey