We teach students to "do" math, not to understand it.by Thomas Fowler-Finn on 03/08/15
Conceptual understanding of math is rarely found in American classrooms. Math is most often taught (regardless of the level of technology used) as content over which students are expected to gain command by applying the correct procedures to get the right answers. Too many students graduate by scoring well enough to pass tests but with little understanding of math concepts. Students leave us with a fear of taking math in college because they know they do not understand the content.
In a recent instructional rounds session I ("tff" in the dialogue below) observed in a 7th grade math class in which each student had their own computer working at their own pace on their own unique math problems. This classroom is in a new school of technology in a school district committed to accelerating learning by using cutting edge hardware and software. I conversed with several students as they worked. The conversation with one boy in particular (identified as "s" below) was revealing and typical of many observations and conversations I've had in schools across the country in which conceptual understanding is a missing element:
tff: "So what are you working on?"
s: "My computer is still loading, but it's a worksheet."
[computer screen now shows: "How is the graph of 'g' derived from the graph of 'f ' ? - answer without doing any graphing." There are 4 multiple choice answers.]
tff: "How will you solve this?"
s: "I have no idea."
tff: "So what are you going to do?"
s: "I'm going to guess and see what happens." [s starts with guessing "A" and the computer message is: "Go back & change your notes. Rework the problem."]
tff: "So did the computer message help you?"
s: "No, not at all."
tff: "So now what?"
s: "I'm going to guess again."
tff: "If you don't know what else to do, guessing is one place to start."
s: Shakes his head in agreement and guesses "C" [The computer message is: "Nice work!"]
The student and I look at each other. He laughs, and I can't help but laugh with him.
tff: "So what is your next step?"
s: "I still don't understand it, so I'm going to pull up a video." [The computer screen shows written step-by-step directions on a procedure to arrive at the correct answer. He carefully reads and tries to rework what he has on his paper.]
tff: "So now do you understand it?"
s: "Not really."
The end of class is now drawing near, and students begin packing up...
tff: "If you are given another one of these problems, what is the likelihood that you will be able to solve it?"
s: "I think it's about 50-50."
Accomplishing conceptual understanding requires restructured lessons with authentic applications that take more classroom time to teach...time that is often hard to come by because of the pacing and testing schedules. The necessary lessons must be structured and guided by teachers who themselves have a conceptual understanding of the content. And just as a doctor cannot diagnose a condition and help a patient get well unless the doctor ascertains the patient's symptoms and pains, teachers must seek out misunderstandings and misconceptions of each individual student in order to bring about their understanding and learning.
This is the first of several blogs over the coming months that will explore procedural versus conceptual understanding.