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The View from Behind the Counter (John Dewey)

Regular Edspresso readers know "John Dewey" is working towards certification as a math teacher.  Click for his first, second, third, fourth, fifth, sixth, seventh and eighth columns.  As always, he prefers to remain anonymous. -ed

Exalted Readers:

Greetings and thanks to my many fans and well-wishers for their undying support, encouragement, wisdom and guidance.  I am happy to say that my Math Teaching Methods, Part I is at long last over.  For those of you wondering how I’ve done, I’m getting an A in the course.  I have not kept secret from the teacher my opinions of how math should be taught and though we disagree, he has offered me this final email message: “I have very much enjoyed sharing the classroom with you.  Your insights and comments have been extremely valuable, and your willingness to communicate your point of view has served as model behavior for your classmates.  Thank you very much.”

There are some positive aspects to Mr. NCTM I’d like to mention.  He has had 30 years of experience teaching high school math, knows quite a bit of math, has a good sense of humor, and has provided my class excellent advice regarding classroom management issues, and other things such as how much material to cover in one lesson plan, and what concepts students find difficult.  Our difference in opinions has not influenced the grading of any of my work.  (Note: He does not yet know about this column, so if you wish to tell him about it, please wait until after the grade is in the transcript.)

My classmates are quite bright, and if I led you to believe they are all dyed-in-the-wool constructivists, let me set the record straight.  Only one is gung-ho, three or four are willing to give it a go, and the rest keep silent.  The young man who is contrarian and with whom I got into arguments is the son of a mathematician, and is quite demanding of rigor.  I suspect that as he gets older his love of rigor will point him in a direction quite opposite that of constructivism.

So where does all this leave me and Mr. NCTM?  Right where you found us.  My idea of “discovery” is quite different than his.  Take the problem of finding a formula for the sum of interior angles of a convex polygon which we discussed in class.  The solution hinges on the fact that the sum of the angles of a triangle = 180 degrees.  You could approach teaching this lesson by guided discovery, and show how to split up a quadrilateral and pentagon into triangles to derive the sum of the interior angles.  After several minutes of discussion, some students may identify key patterns and the teacher could wrap it up.  I suggested this to Mr. NCTM, remarking that it really wasn’t giving away the store and there was still some discovery involved.  

He nodded acknowledgement but continued to “guide” our class to the constructivist approach: 45 minutes of discovery including having students actually measure the angles of various convex polygons with a protractor, and after accounting for error in measurement, making conjectures and seeing “patterns”.  One fellow student asked why one would do that when in fact geometry was about deductive reasoning and learning to reach conclusions about measurement without the aid of actual measurement devices.  Mr. NCTM said students should be given free reign to discover the superiority of the deductive method.

For fear of being forever branded as a blog poseur, I hesitate to say whether my approach would be called “direct instruction” or “guided discovery” or Vygotsky’s Zone of Proximal Development and its related term, “scaffolding.”  I just offer that the type of discovery my approach entails is what I and others in my age group had growing up. Given a choice between giving students 45 minutes to reach an “aha” experience, or 5 to 10 minutes, I and others like me opt for the latter.

It is interesting how one imbued with the NCTM philosophy of teaching views the world.  In a discussion about textbooks, we asked him what books were his favorites. To my surprise, he liked Foerster’s algebra, and Jacobs’ geometry texts—two classic and very traditional texts.  Nevertheless, as good as they are, in the NCTM view (and Dr. Cangelosi’s as well–see my seventh missive), textbooks are mere resources by which to provide exercises and problems for the “lower level” algorithmic skills.  The real teaching and learning is in constructivist lessons.  In answer to why there are bad textbooks out there, Mr. NCTM remarked that the NCTM standards haven’t made their way into the textbooks as much as he would have hoped.  That the NCTM standards themselves may be a reason for bad textbooks is probably not something he has even remotely considered—just a guess.

Which brings me back to the drugstore in a scary part of town where I once worked (also talked about in letter #7).  Something else happened there that is relevant to the differences between us.  While working the day shift, I was given instruction by a seasoned veteran on how to operate the cash register and not take any guff from customers.  The customers were the enemy in her eyes—not to be trusted, not to be friends with. We ran the show. 

Because I was also given night shift duties on occasion, my father was not thrilled with me working there, and I was forced to find another job.  About a week after I left the drugstore job, I returned as a customer.  I saw my old trainer/mentor behind the counter and I waved to her.  Her eyes passed over me like they did everyone else on our side of the counter. I was now the enemy, unrecognizable as someone she once counseled, and on the other side of a gap that would never be bridged.  I am grateful to both teachers and wish them well.  But I know on which side of the counter I will remain.

My best wishes for the holidays.

John Dewey


  1. Michael Paul Goldenberg says:

    I see you lacked the intellectual honesty to post the critique I sent of this column. And you have the gall to post one that suggests that NCTM-supporters are the ones who don’t tolerate diversity of thought? What a shameless coward you are. Happy New Year. Fox News awaits your services.

  2. John Dewey says:

    Thanks to Dave Marain for your thoughtful comments. Future columns will continue to address the issue of constructivism in its various incarnations, and the people who love it, as well as those who don’t. Happy new year to all!

  3. Dave Marain says:

    Happy Holidays Mr. Dewey!
    I invite you and your faithful to read my expanded comments in my ‘newish’ blog http://www/mathnotations.blogspot.com.
    At any rate, I’ve just become aware of your excellent writings and I commend you for your thoughtful insights. Ok, enough of the brownnosing… Here’s the skinny. I believe you’re on the right track with your thoughts about constructivism. My comments regarding this appeared in Joanne Jacob’s blog so I won’t go into detail here. As children move up the ladder, they should need less of the hands-on manipulatives and tactile experiences like the one described in my post. This is why I suggested that cutting off the corners from the vertices of a polygon is a worthwhile activity for FOURTH graders, but I would not spend that kind of time for middle schoolers. Once they’ve experienced the hands-on approach, they can quickly revisit this for a triangle, then move onto a more abstract pattern-based approach in the middle grades, e.g., dividing a polygon of n sides into n-2 triangles (they can formulate this for themselves within 5 minutes), thereby developing the standard formula. By the time they reach a traditional geometry course in hs, they’ve had the spatial experience from 4th grade, the pattern-based approach in middle school and therefore they can quickly review this and focus on APPLYING the formula to regular polygons and more sophisticated algebraic exercises. But this discussion has important implications:
    0. None of my remarks makes any sense unless 4th, 8th and 10th graders in downtown Chicago are exposed to the same learnings as those in the affluent suburbs of Chicago off Lake Michigan, if you get my point. THERE MUST BE ONE NATIONAL MATH CURRICULUM and it cannot represent one side or the other in the Math Wars. Your approach is a good one, Mr. Dewey, because it is a blend, but you might need a bit more field experience before deducing general principles. I’m not being patronizing or condescending here, so pls don’t take it that way. Radical solutions from either camp can be dismissed but how we combine the best of traditional and reformed approaches is not so obvious.
    1. More hands-on in lower grades (assign any label you want!) with teachers who are committed to this and properly trained
    2. Gradual development of abstract formulations of patterns with algebraic representations starting much earlier in Middle School than is the norm in the USA and ONE of the reasons why we lag behind other nations.
    3. More challenging applied problems for the hs students instead of merely rehasing formulas and doing the standard problems
    FINALLY, SOME APHORISMS (mock me if you will!):
    4. Despite ‘cutting the corners’ for polygons, THERE ARE NO SHORTCUTS FOR DEVELOPING A PROFOUND UNDERSTANDING OF FUNDAMENTAL MATHEMATICS! Here’s what I tell my students and they don’t think the reference to their grades is amusing:
    The only shortcut in math is from ‘A’ to ‘F’! [ok, you can groan loudly now!]
    Dave M

  4. Niki Hayes says:

    Mr. Dewey will provide, I hope, a complete service by printing this blog info in a book someday. Talk about qualitative research…

    My favorite sentence is the following:
    “It is interesting how one imbued with the NCTM philosophy of teaching views the world.”

    It is, to me, a case of NCTM people being from Mars and traditionalists (classicists?) being from Venus. There is no room for diversity in thought on Mars. Those of us on Venus really do accept conceptual learning; we just want full citizenship and participation on Mars!

  5. Wayne Bishop says:

    “In answer to why there are bad textbooks out there, Mr. NCTM remarked that the NCTM standards haven’t made their way into the textbooks as much as he would have hoped.”

    Was he asked and, if so, did he offer examples of NCTM Standards-consistent textbooks that he believes are good? Such as Connected Mathematics in middle school?

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