As a first year teacher, I am not sure how good I was, but I was certainly enthusiastic. I had discovered a talent for mathematics while at college, and a passion for interacting with young adults even earlier. But my first attempts at combining the two were necessarily clumsy.
In math classrooms, it is sadly too common for students to view the material in a passive, receptive way. Mathematics, to many, is a foreign language, or perhaps a foreign religion. You can, through trial and error, memorization and repetition, or the perception of social cues, start to guess what the priest of mathematics at the chalkboard wants you to do or say. Sometimes the high priests of mathematics will pass on answers to the odd questions so you can see if you are wrong or right, but you don’t imagine that you are supposed to understand each step–that every line is a choice, and that every choice must be valid and should be helpful. You know things are right or wrong, but you don’t spend too much time worrying about if they make sense.
In my first few weeks, students rarely asked questions, and never corrected me. I was confused because I am fairly inconsistent with arithmetic, and I know that I had always loved nothing more than pointing out to my teachers some mistake they hadn’t caught. Slowly, it dawned on me that my students were not in the habit of questioning the teachers mathematical authority. I was the priest, they were the catechumens (and they were more or less expecting the mathematical equivalent of the Baltimore Catechism).
In an attempt to remedy this, I stole an idea from one of my Professors. I hung a crude poster next to the board and explained to each of my classes that every time I made a mistake and it was caught, by a student, I would mark a tally under their periods section of the poster. When the class had reached 10 tally’s , I would buy them donuts.
Market-obsessed education reformers go on and on about incentives, but rarely bother to ask what kind of behavior they are trying to promote. With Donut points, I did not reward students for scores or for grades, but for a simple behavior (calling me out) which, I believed, would have a huge impact on how students engaged with mathematics and viewed themselves as students. I wanted to give them freedom and responsibility, and, after some tweaks to the policy (no shouting, no crying, no changing my work and trying to set me up) it was very successful.
As the year progressed, we had many fruitful discussions about whether a particular mathematical statement was a mistake or not. If it was, what type? Should it count? Does a misspelling count? Does a poorly drawn graph count? Does a hardly visibly minus sign count? If these things count, how do they get treated on exams? I can’t tell you how “proficient” this made everyone in standard x,y,z but I am confident that my students learned, albeit slowly, that they had the potential to speak and therefore be mathematical thinkers. I’m not sure it was enough, but it was a start.