Out in Left Field proudly presents the fifteenth in a series of letters by an aspiring math teacher formerly known as "John Dewey." All personal and place names have been changed to protect privacy.
With all the scandals about teachers and administrators cheating to raise their students’ scores on standardized tests, Miss Katharine was a bit concerned over my letter of a few weeks ago in which I described how I handled an assessment I had to administer. I told her she didn’t have to worry: no one was out looking for me, and the only knock at my front door was from a young man who was selling magazine subscriptions that would serve some worthy cause which had benefitted him somehow and which slips my mind at the moment.
Recently I was back at the same school where I had administered that Common Core flavored exam and ran into one of the students from the 8th grade algebra class. They were the ones who had demanded to know what “explain your reasoning” meant.
“Oh, you’re the guy who gave us that explanation for how to do the problem that no one could do,” she said. Interesting that the explanation I gave took less than a minute but apparently was enough to get the point across; so much so that I’ve become somewhat of a legendary sub. Which perhaps suggests that explicit and direct instruction might not be the “rote learning” approach feared by otherwise sane and pleasant people.
The “understanding” and “connection” mantras are prevalent in the groupthink that makes up much of the education establishment’s view of math education. It was at play big time with the algebra program I had to use when I was student teaching—a program called CPM algebra. I’ve mentioned it before and it seems appropriate to mention it again what with Common Core surfacing and being interpreted along the ideologies of reform math.
With CPM algebra, students were taught “slope” in a series of discovery lessons that spanned many weeks. They had to make “connections” between tables of values, and equations, how both described the patterns of growth, how the y interecept value helped to draw a graph, how to draw a graph given an equation, and how to determine the slope and y intercept when looking at a graph. All well and good, but the point-slope form of finding an equation was not presented initially; students were left in the dark for quite a while about how to determine the equation of a line given the coordinates of any two points on the line.
I kept to the script of the algebra text as best I could. This turned out to be disturbingly easy. You just went over the previous problems, gave a short intro for the topic of the day to get them going and then assigned the problems in the book for that day. They then worked on them in groups and Tina and I circulated to answer questions. I could see how if a teacher were lazy (which I hasten to say Tina, my supervising teacher, was not), they wouldn’t have to do very much, and lesson plans were pretty much automatic. Tina bought into the program; she believed in it and worked hard to make it work. But it also seemed she was seeing what she wanted to see. There were times when, circulating around the classroom, she would say to me “They’re getting it! They’re making connections!” Yet, there were students who seemed quite confused and some of them knew that if they pushed me hard enough during my circulatory tour of the classroom, my hints (given while Tina was working with other students and out of earshot) would often tell them what they were supposed to discover. Maybe the connections she was seeing the students make were because of that.
The reason why CPM eschews procedures like the point-slope method of finding an equation is that it supposedly gets in the way of true understanding. I heard this recently from a teacher, in fact: “Kids buy into the slope formula, plug in numbers, do the calculations and yet they still do not understand what they are doing. They are simply memorizing yet another formula for some unknown reason.”
I don’t know. I just don’t find slope all that terribly difficult to understand. Similar triangles and proportion seem to explain why the slope of a straight line is always going to be the same for any two points you pick. But people seem to think that if a kid is doing procedures without “complete and true understanding” he's doomed to a life of failure. It is as if the moment a student stops doing all the intermediate steps/algorithms and fails to make the appropriate connections each time, then he or she is using a trick or rote memorization to jump to the end result and not using understanding or strategies to solve something.
I recall one time when student teaching, talking to a fellow math teacher. This was during the time that Tina was gone for two weeks when her father passed away. The teacher was telling me about the math teaching philosophy. “Tina always says we can teach them how, but what’s really important is that they understand ‘why’". As she told me that, she looked to me as if she wanted me to say something. I sensed that underneath it all, she felt the same way I did—but was afraid of being disloyal.
I think of that hallway conversation often. I think of it when I see the posters for the Standards for Mathematical Practice on the walls of the various classrooms in which I substitute. They make me feel as if I’m back as a student teacher, trying to figure out the best way through a ridiculous program. And despite my strong beliefs about what I talk about here, I still feel like I’m cheating when I teach the way I see fit, as if maybe 1) there's something wrong with me, or 2) I’m being disloyal. I’ve only met a few teachers who have told me they don’t like the trends I’ve been describing in math education. They’ve usually been teaching for over 30 years and are about to retire.