What We Say, What Our Students Hear: A Case for Active Listening

Abstract

I want us to think about what our students hear, which is often not what we are trying to convey. I think we can all believe that things go on in our students' heads that we don't understand and that we need to pay more attention to. I suspect that more than one of you has put a problem on a test that lots of students have answered incorrectly and you've said, " How could they mess that up? " I hope that this paper will give you some clues about why they " messed that up. " More importantly, I want to encourage you (and me) to listen more carefully to what our students do say. The " active listening " in this paper involves listening on OUR parts. What are some messages that our students hear? We say, " This won't be on the exam. " They hear, " This is not important. " We say, " You will need this concept next year. " They hear, " I don't need to learn this concept this year. " We say, " We want you to use algorithms quickly and automatically. " They hear, " Mathematics does not require thought. " We give timed tests. They hear, " Mathematics must be done quickly. " They, therefore, will not struggle with problems that they cannot complete quickly. We give them lots of exercises with no words. They hear, " Mathematics is not a language of communication , only computation. " We don't give partial credit. They hear, " The mathematics is the final result, not the process. Mathematics is either all right or all wrong; there is no middle ground. " ONE MODE OF REASONING To begin to understand more deeply what our students hear, I want to think about mathematics, and about one suggested style of reasoning that people might use in mathematics. • Gets right to solution in a structured, algorithrmic way, stripping away any context. • Uses a mode of thinking that is abstract and formal. • Geared to arriving at an objectively fair or just solution upon which all rational persons can agree. • Employs a legal elaboration of rules and fair procedures. • Confident to judge. • Is analytic. How does this reasoning style relate to mathematics? Think for a moment about this reasoning style. Does it describe mathematics for you? Do you think it …