My step-son, Sam, is one of those otherwise bright students who struggles with math. Back when he was in high school, his mom asked me to help him. He had gotten a question wrong on a Geometry quiz and didn’t understand the correct answer. My wife hoped that since I was a former high school math teacher that I could help him out.
The question was, “What is the intersection of two planes?”
He told me that he had answered that the intersection was a point, since only lines intersect. Sam went on, “I went in to ask my teacher about the question, but she just kept giving me the right answer. I really don’t understand it at all.”
“So, you’ve only talked about lines intersecting?”
“And you haven’t really talked at all about planes and how they intersect?”
Sam shook his head.
“Then I could see why you thought it was a point,” I told him. “But look at this.” His notebook was on the kitchen counter where we were talking and I said, “Let’s say this is one of the planes,” while tapping his notebook. I grabbed a magazine, saying it was the other plane. I held the spine of the magazine at an angle against the face of Sam’s notebook.
“How do these two planes come together? What kind of geometric shape?” I asked.
Sam got one of those “Oh, my gosh! Is it that simple?!” looks on his face and said it was a line.
Now, there was nothing wrong with the teacher asking the plane intersection question without first modeling it for students. It is a great way to have students apply the concept of intersection of geometric shapes and see if they really understand it. And the teacher was a kind and knowledgeable math teacher.
But students who struggle with a subject need more than just someone who is sensitive and kind and knowledgeable. Sam needed more than the correct answer. I think teachers who are intuitive mathematicians (or social scientists, or literacy specialists, or scientists) know their subjects in an intuitive way that makes it hard for them to explain ideas to students who do not understand their subject intuitively.
When students get an incorrect answer, it is too easy for teachers who understands their content intuitively to assume that the student simply made a mistake (perhaps in calculating), or didn’t study hard enough, or is simply a slow student in their subject.
What they don’t understand is that more often than not, a student’s wrong answer is actually a correct answer within the student’s current (but incorrect) schema for the topic – the student’s internal model that tells him how things work.
If the teacher’s goal is to have the student understand the material correctly, then simply offering the correct answer is less productive than trying to understand the student’s misconception and then think of an example or a way to model the material that will create a bridge between the student’s misunderstanding and the correct understanding.
Sam’s schema said only lines intersect and he knew that lines intersect in a point. We could either stop with proving that Sam was wrong by giving him the correct answer, or we could work to understand his thinking so we could lead him in the right direction.
I don’t blame the teacher. She simply did what I did when I was a math teacher. It wasn’t until long after I stopped teaching math and became of student of learning that I grew to understand this principle.
How much more effective would our teaching be if we approached our students’ incorrect answers as misconceptions rather than missing information?