Kim Mackey wrote: > > In my readings on constructivism, email conversations with various > psychologists, and discussions on this newsgroup, it seems that every time I > think I am getting close to coming to grips with a constructivist classroom > I find myself confronted with smoke and mirrors. If everyone is a > constructivist, what distinguishes it from traditional practice? I am not > speaking now of the _caricature_ of traditional practice, but traditional > practice as done by teachers who are well-versed in their subject matter and > who incorporate a variety of strategies and techniques in their teaching > style. Let me expose some of my techniques and specify which I think are > constructivist and which I think are not. This way we have a target to shoot at. > > Constructivist Practices: > ============================== > Problem of the Week write-ups. > > Quarterly Portfolio of written work with a page of self-assessment > > student presentation at the board of their attempt to solve problems > introducing new concepts > > cooperative group work on non-routine solid geometry problems > > continual emphasis on reasonableness of answers > ====================================== > Non-constructivist practices > ====================================== > > students required to memorize 19 formulas over a year's time for mental math > > students required to memorize the pythagorean theorem, vertical angle > theorem, and the 180 theorem > > continuous practice of mixed problem sets over the course of a quarter or > semester > > explicit teaching of concepts > > use of mnemonics (SOHCAHTOA, PEMDAS) > > 70 percent of grade by examination each quarter, 30 percent quizzes (6 a > quarter), 40 percent tests (3 a quarter). > > feedback to students to help eliminate procedural bugs > ============================================================================= > > So what do the constructivists in the audience think? > > regards, Kim Mackey
I think constructivists in math education for the most part (myself among them) believe that acceptance of cognitive constructivism (all knowledge is constructed by the learner, etc.) makes aceeptance of certain pedagogical stances imperative--i.e. learning is an active, dynamic process, knowing is doing, etc. I think what most Constructivists object to in a classroom dominated by direct instruction is the overwhelming potential in such an environment for ignoring the necessity of making the learner an active participant. All knowledge is constructed, but as any builder will tell you not all constructions are equal. Active learning and practices recommended by the Standards and other reform documents are intended to help learners build "strong" contructions. Memorizing formulas, drill & practice, etc. tends to work in the short run, but soon the "wheels fall off the cart" and we are back at square one. For example, I teach a class of "pre-Algebra" to 7th graders. They know lots of stuff about "mathematics" and have memorized many procedures and facts. When asked to evaluate: 2/3 + 2/3 One responded: "I got 1. You add the bottom 3 from the first and the top 2 from the second and then put that over, like the top number from the first plus the bottom number from the second one, so it's 5 over 5 which is one." A third of the class agreed. Another student said, no that can't be right because "2/3 and 1/3 is 1 and we're adding more than 1/3." "Oh, replied the first student, well I know you do that top and bottom thing with fractions sometimes." None of the students could create a representation or picture of the problem. I suppose many would say I should have just said, "You know how to do this, you just add the numerators and put them over the common denominator." I didn't and after some work with representing things pictorially, they told me, "Oh yeah, I remember now." and proceeded to describe the correct algorithm.
They had indeed constructed some knowledge, but most of what they have is weak and focuses on "how to", they often misremember and since the constructions lack depth and strength they cannot "dig out" If they don't remember the specific procedure or algorithm, they are stuck.
P.S. These students have spent most of their prior time in mathematics classes drilling and practicing, listening to "explanations" of "how to" from dedicated teachers who have well-managed classrooms in which students meet success and can pass the chapter tests. Honest.