This is the eighth 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 and seventh columns. As always, he prefers to remain anonymous. -ed.
After last week’s missive quoting from Dr. Cangelosi’s textbook, I expected he would have left a comment expressing his eternal gratefulness for the exposure I gave his book. But he lives in Utah, where the state legislature there recently adopted a resolution that calls for the Utah State Board of Education to give Utah’s math standards an overhaul. That may have him worried and I’m sure that’s why he hasn’t written.
From what I see and hear in ed school, Dr. Cangelosi doesn’t have a thing to worry about. The milieu-controlled ed school environment of discovery/inquiry based/NCTM-standards-based/constructivist-based/brain-based/knowledge-based/critical thinking-based/ and higher order thinking skills-based learning is ever expanding.
Indicative of this brave new world is a comment that Mr. NCTM left on a lesson plan I turned in—an assignment that called for a lesson which made use of technology.
My lesson plan had students explore the graphs of quadratic equations using graphing calculators. I borrowed heavily from exercises in a math book by Gelfand on functions and graphs. In one of Gelfand’s non-calculator based exercises, he asks the students to graph y = x2, and then y = 5x2 and asks “What scale unit would have to be taken along the axes in order that the curve for y = x2 could serve as a graph of the function y = 5x2?” Students are to provide a rule linking the shape of curve to the x coefficient, based on their answer to the scale unit question. Mr. NCTM wrote in the margin: “This is just the kind of ‘discovery’ learning that you have rebelled against.”
His comment reminded me of the movies made in the Cold War era in which a staunch Soviet leader says to the American hero: “Perhaps, comrade, we are not so far apart as we thought.” I believe what he saw in my lesson plan was my clever camouflaging of what is called “scaffolding” as “discovery”. Scaffolding refers to the providing of information and knowledge to allow students to apply such knowledge to a new situation or problem. So perhaps in this sense he is correct that we are not so far apart. But in other areas, despite the optimistic nature of his remark, I do not feel we are really as close to glasnost as he would like me to believe. Perhaps he sees my clever camouflage as a chink in the armor on the way to get me to see things the NCTM way.
His view of me as a dissident in need of enlightenment comes from things I say in class, most recently in a class discussion on the role of graphing calculators in math education. The discussion started in the usual manner: get in small groups. In my group was the fellow with whom I had an argument about state standards the first class and which I talked about here. I’ve grown to like him; he’s very young and full of opinions and enjoys being contrarian. For many people in their twenties being contrarian is a quest for identity until marriage, work and humility take over, and not necessarily in that order. In any event, for this young man none of these things have yet kicked in.
Mr. NCTM facilitated the classroom discussion. While we are not Luddites in our class, and can appreciate the value of graphing calculators in teaching, we also saw problems. After several minutes the whiteboard was filled with issues including overdependence, obscuration of concept, and interference with conceptual mastery. After some discussion of the pros and cons of graphing calculators, Mr. NCTM decided to change tack on us and asked: “Do you think they are introduced too early?” (They are introduced as early as kindergarten in some programs.)
Our answers were going in the direction of “yes”, until my young contrarian friend spoke up and said, to Mr. NCTM’s obvious delight, that he really couldn’t see what was the cognitive value of teaching students the procedure for multiplying 36 x 7 when calculators were available. I was unable to keep my mouth shut. “Don’t you think that students need an understanding of basic procedures and that place value is an important concept?” “Why?” he remarked and went on to the uselessness of learning long division at which I drew the line and said “How can you say that? Don’t you think the distributive property is worth talking about?”
“Who cares?” he pointed out.
Mr. NCTM was enjoying this debate immensely. Dialogues such as these apparently feed into his fantasy that he’s actually teaching in a real grad school in a real program.
Mr. NCTM took over and allowed that there was some value in teaching the long division algorithm and perhaps some value in multiplication algorithms, but after that, it is just so much tedium. “There are some who feel there should be no pencil and paper calculators in classrooms at all; you either do it in your head using estimation or you use the calculator. It breaks my heart when I see kids writing down 32 divided by 2 and solving it as a long division problem.” It breaks my heart too. Students used to be required to practice problems such as these until they could do it in their head as he would like to see. Such problems used to be called “short division”. Apparently, Mr. NCTM sees long division as causing this problem, not the calculator.
“Let’s put it this way,” Mr. NCTM said. “If I saw a student who was not able to perform the division problem of 168,514 divided by 384, that would not be a reason for me to hold him back from taking algebra.” Well, if it were me, I would first want to know why he couldn’t do the division problem and then what else he couldn’t do.
Which tells me that Mr. NCTM and I are a long way from perestroika.
From the gulag of math teaching methods, I remain,
Faithfully yours,
John Dewey