What I Learned from Rusty
By Dr. Eugene Maier
When I was in school, I decided that an answer to a mathematical question wasn't complete unless I could explain how I got it. I had plenty of evidence to support this belief: Docked points on papers, with "Show your work!" written alongside, and the inevitable "How did you get that answer?" when called on to recite in class. Clearly, an answer by itself wasn't enough.It didn't matter if the answer was obvious to me, I had to make it obvious to the teacher, too. Sometimes, given the way the teacher thought, or if I wasn't exactly sure where my idea came from, it was better to keep quiet. When doing school math, knowing the answer wasn't enough.
I carried this belief into my own classrooms, passing on to my students the message ingrained in me by my teachers: correct answers aren't sufficient; you must also explain how you arrived at them.
And then I met Rusty. Rusty was a fifth grader. I was involved in project SEED. I don't remember what the acronym stood for, but it was an organized effort to get professional mathematicians and scientists into elementary classrooms, especially in low-income areas. So each day, for one school year, I left my office at the University and drove to the other side of the tracks where I spent 45 minutes doing mathematics with Rusty's class. They still had their regular arithmetic period outside the time I spent with them, so I was free to roam over a variety of topics.
Rusty's desk was in the back of the classroom and, despite classroom distractions and his apparent inattention, Rusty, I discovered, was following whatever I was presenting. As I posed problems for the class to work on and roamed about checking on the students' progress, Rusty was among the first to arrive at an answer. This surprised me until I learned that his way of operating gave little indication of his involvement. And whenever I asked him how he had arrived at his answer, his usual response was a shrug. Pushing for a further explanation elicited nothing more than some variation of "I just know" or "That's what it has to be."
I was struck by Rusty's confidence in his answers and how unfazed he was by my queries. My university students--better versed in the nuances of professors' comments--might well have become suspicious that there was something amiss in their thinking or that I had asked them a trick question.
I puzzled about the course I should take. I could have followed my usual path and kept after Rusty to describe how he was arriving at his answers, suggesting to him that unless he did so, his answers were suspect. But I was convinced that Rusty had no doubts about his answers. I knew they were correct--what I didn't know was how he arrived at them and asking him about his thinking was futile. So I decided to accept Rusty's answers without question. If I had doubts about their authenticity, rather than questioning him about how he arrived at his conclusions, I would change the parameters of the problem. If Rusty adapted his answer accordingly, that would be evidence enough for me that he knew what was going on.
How I decided on this course of action is, to some extent, as mysterious to me as how Rusty arrived at his conclusions. Somehow it dawned on me that there was a difference between knowing something and explaining how one comes to know it; one could know something and not have the verbal skills or the conscious awareness of one's mental processes to be able to explain how that knowledge was acquired. That seems obvious to me now, but at the time knowing the answer to a problem in school mathematics and being able to explain how one reached it were so interwoven by years of external and internal messages that separating them required a mental shake-up.
Also, I was struck by the contrast between Rusty's quiet confidence and the trepidation of some of my university students. I had students ask me how to solve a problem only to discover they had reached a correct solution but believed it invalid because they hadn't used a prescribed school method. They didn't trust their natural mathematical insights and intuition. I decided I didn't want that for Rusty--I suspected his solutions sprang from a keen, intuitive number sense. I didn't want him to lose trust in or abandon it, and I didn't want him to believe his thinking was suspect or unacceptable because he was unwilling or unable to describe it.
I questioned why so much emphasis was placed on showing work and explaining thinking. If one develops the ability and confidence to deal with whatever mathematics arises in one's life, what need is there to explain how one does that? I had no doubt that Rusty would do just fine dealing with whatever mathematics came into his life. His verbal skills might be lacking, but that was another matter. I didn't want to see his mathematical intuition get discounted or, worse yet, destroyed.
As a result of my experience with Rusty, I quit discounting students' results if they couldn't explain how they arrived at them. If I doubted the validity of their methods, I would do what I did with Rusty--change the setting and see if they still got correct results; if not, I would simply report to them that something in their thinking had led them astray. I still asked students to reflect on their thinking. Describing, and listening to others describe, mathematical thought processes can add to one's insight. But thought processes can be elusive and, even if captured, may be difficult to describe in words. But that doesn't mean the quality of one's thinking is inferior or the result of that thinking is somehow inferior. As I learned from Rusty, there's a difference between doing mathematics well and describing how one does it.