I have mentioned in earlier posts, some of the problems with the way students typically view themselves in relation to mathematics and mathematicians. This is a deep problem, which pops up throughout our culture.
Let me illustrate: How often have you heard an adult say (or said yourself) to a kid, “You don’t like (or are struggling with) math class, oh that’s ok, I am not a math person”? Have you ever heard someone make a parallel statement about reading…”Oh me too, I’m not one of those people who reads”. Is it surprising that students think math is inaccessible and reserved for the elite.
Even those who view themselves as good at math are spending the majority of the time repeating processes and rules that they only partially understand (which would be alright if they were also being encouraged to ask questions about those partial understandings). Students don’t speak about mathematics. Teachers speak about mathematics, textbooks speak about mathematics, the state and the principal might speak a little about mathematics. These actors have authority. Go to any classroom (mine included) and you will almost certainly find a version of a pattern in which a teacher asks a question (which presumably he or she already knows the answer to), a student answers the question, and the teacher pronounces the answer right or wrong. Sometimes the textbook takes the place of the teacher as the pronouncer. But isn’t the beauty of mathematics that the right answer is right without pronunciation? Aren’t their deeper ideas to be explored? Do students and teachers have a sense that many of the things we declare right or wrong in mathematics are actually choices (negative times negative is positive, for example)?
Its not always bad to tell students what is correct and incorrect, or to ask questions you already know the answer to, but it is bad to always do these things. It creates a culture where you do mathematics they way you do it because that’s what you were told to do. Then we all wring our hands with surprise when students are bored, angry, alienated, or simply defeated by the exhausting list of procedures and rules. Standardized tests obviously accelerate the problem since the least risky way to prepare for these tests is to compartmentalize each type of question and then try to execute the correct procedure when that question comes up.
In what is sadly a very typical exchange, I was observing a classroom where a student showed me her attempt to add fractions. In her work, she had simply skipped the common denominator, adding “tops” and “bottoms”, a mistake that quite common. I am not sure if she sensed from my face that something was out of place, but she quickly stated, “that’s not right is it”? I broke the mold a little by asking her why she thought it wasn’t right, to which she replied innocently, “because that’s how I do it on tests and they always tell me that I am wrong”. This was not a young girl, but a student who had probably been receiving lessons on this topic for years. Its clear that she did not feel she knew mathematically why her work was wrong, or that she had any sense that this type of knowledge might be important.
In my research, I have worked with teachers who want to try to address these types of problems. The good news is that there are lots of things to try. The bad news is that none of them work very well. It is very difficult to overcome an entire culture. Students can get nervous, teachers can get nervous, parents can get nervous, administrators can get nervous. And if your test scores should happen to go down, then what? Still, I would urge any math teachers to think about these difficulties. We should expose and consider the problems in our practice. How can we ensure that our classrooms are places where teachers and students do mathematics? How can we promote and encourage our students to have a sense of control and authority in their mathematics education?
If you have anything good, let me know.