Precise and thoughtful communication supports both correctness and clarity.
In Making Sense of Algebra, Paul Goldenberg, June Mark, and their colleagues look carefully at how our students think about mathematics. They explore five “Habits of Mind” that focus not just on the results of mathematical thinking, but on how proficient students do that thinking.
In today’s blog, which is the last one in a series of five adapted from the book, the authors talk about the habit of communicating with precision.
By Paul Goldenberg, June Mark, Jane Kang, Mary Fries, Cynthia Carter, and Tracy Cordner
The habit of striving for clarity, simplicity, and precision in both speech and writing is of great value in any field. In casual language, we are used to using context and people’s reasonable expectations to disambiguate communication so that we don’t burden our communication with specifics and details that the reader/listener probably can be expected to surmise anyway. But in mathematics, we need to base each new idea logically on earlier ones; to do so “safely,” we must not leave room for ambiguity. Mathematics, in which such precision in communication is essential, is a good training ground for clarity. Communication is hard; it takes years to learn how to be precise and clear, and the skill often eludes even highly educated adults. If the teacher and curriculum serve as the “native speakers” of clear mathematics, students can learn the language from them.
Defining terms is important, but children (and, to a lesser extent, even adults) almost never learn new words effectively from definitions. Virtually all vocabulary is acquired from use in context. Children build their own working definitions based on their initial experiences. Over time, as they hear and use these words in other contexts, they refine working definitions, making them more precise. For example, baby might first use dada for all men, and only later for one specific man. In mathematics, too, students can work with ideas without having started with a precise definition. With experience, the concepts become more precise, and the vocabulary with which we name the concepts can, accordingly, carry more precise meanings.
For learners, precision is also a way to come to understanding. By forcing their insight into precise language (natural language or the symbols of mathematics), they come to understand it better themselves.
For teachers in a class, this kind of clarity is important so that students understand exactly what a question means. Incorrect answers in class, or dead zones when students seem baffled and can’t answer at all, are often the result of not being sure what question was asked, rather than not understanding how to answer it.
Clarity comes from developing a variety of subskills:
Clear communication requires the refinement of academic language as students explain their reasoning and solutions. Along with some new, specifically mathematical vocabulary, this includes the use of quantifiers (all, some, always, sometimes, never, any, for each, only, etc.), combination and negation (not, or, and), and conditionals (if…then…, whenever, if not, etc.).
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Click here to learn more about Making Sense of Algebra.