Communication in a Mathematics Classroom

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One of the most notable, powerful, and observable changes that has taken place in the teaching of mathematics (and all other subjects) has been the realization that people learn best in community. Experts now know that the traditional mathematics class, in which each student worked in isolation, actually made it more difficult for students to learn. Recent brain research has helped to confirm that people learn best when they learn in relationship with others. This leads to a vision of an ideal mathematics class in which students spend most of their time in communication with each other and with the teacher. Students are continuously speaking and listening, writing and reading mathematics. Both sides of the communication system—both expressive and receptive—are beneficial to learners of mathematics. Teachers are no longer spending all their time in front of the class telling their students how to do mathematics. Rather, students are doing a lot more of the talking—both in small group settings and in front of the whole class. Writing mathematical explanations, making use of all kinds of graphical and diagrammatic aids, has become an important part of the mathematics learning experience.

Often in the process of learning something new in mathematics, the learner has an idea, or a proposal, or an explanation of a phenomenon that he thinks might be true. For example, students might be exploring the relationship between area and perimeter of rectangles. After noticing that perimeter and area of rectangles can be changed independently of each other, a student might think that this phenomenon could also be true of squares. This idea might occur to the student in an instant of creative thought. In order to engage the rest of the class in testing the idea, that student must find a way to tell his classmates what his idea is. There are any number of ways he might try to express his conjecture. Any attempt to do so will be beneficial to that student. If he succeeds in getting his idea across to others, his classmates can help him explore the idea to find out if it is true. Even if he fails in his attempt to communicate his conjecture, he and his classmates benefit from the attempts. Perhaps as he tries to communicate his proposal, one or more of his classmates will misunderstand the original idea but will think he is suggesting that all shapes have perimeters and areas that can be changed independently. This is not the idea that the first student was proposing, but it is a worthwhile idea to explore nonetheless. In fact, it turns out to be a better conjecture than the original, because it is more general. Mathematicians (and scientists) tend to be greatly interested in relationships that can be generalized. Much learning occurs, even indirectly, as a result of students sharing conjectures and working together to explore them further.

Rich experiences in communicating mathematical ideas must be a part of the learning experience for all mathematics students, including users of this text. It is not uncommon for mathematics learners to say something like this: "I understand it; I just can't explain it." While this may seem like a reasonable and plausible thing to say, it actually could be an excuse for avoiding some difficult thinking. When a learner experiences a flash of insight about a mathematical idea, at first it is possible to "understand" it and be unable to explain it. However, as that person makes the effort to analyze this insight and communicate it, that person's understanding becomes much deeper. Often the first attempts at communicating a new insight are failures. This does not indicate an inability to explain one's thinking. Rather, it indicates a need for more work on the communication. The learner must try another way to get the ideas across. The learner's audience must provide feedback and questions to let the communicator know whether the communication explanation has been clear or not. As a person succeeds in communicating her mathematical insights to others, she comes to a point of deep and meaningful understanding. Without this experience a mathematical insight is a more fragile commodity; it is something that is likely to fade away and escape the learner's grasp.