By June Mark, Paul Goldenberg, and Jane Kang (@EDCtweets), Education Development Center, coauthors of Transition to Algebra.
The word “precision” calls to mind accuracy and correctness. While accuracy in calculation is a part, clarity in communication is the main intent.
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 use context and people’s reasonable expectations to disambiguate meanings so that we don’t burden our communication with specifics and details that the reader/listener can be expected to surmise anyway. But in mathematics, we base each new idea logically on earlier ones; to do so “safely,” we must not leave room for ambiguity.
Defining terms is important, but children (and, to a lesser extent, even adults) rarely 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 words in other contexts, they refine their 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.
Striving for precision is also a way to refine understanding. By forcing an insight into precise language (natural language or mathematical symbols), we come to understand it better ourselves. For example, learners often trip over the order relationships of negative numbers until they find a way to reconcile their new learning (–12 is less than –6) with prior knowledge: 12 is bigger than 6, and –12 is twice –6, both of which pull for a feeling that –12 is the “bigger” number. Having ways to express the two kinds of “bigness” helps distinguish them. Learners could acquire technical vocabulary, like magnitude or absolute value, or could just refer to the greater distance from 0, but being precise about what is “bigger” about –12 helps clarify thinking about what is not bigger.
Here are some ways for teachers and students to develop greater clarity and precision:
Correct use of mathematical terms, symbols, and conventions can always achieve mathematical precision but can also produce speech and writing that is opaque, especially to learners, often to teachers, and sometimes even to mathematicians. “Good mathematical hygiene” therefore requires being absolutely correct, but with the right simplicity of language and lack of ambiguity to maintain both correctness and complete clarity to the intended audience.
♦ ♦ ♦ ♦
June Mark, Paul Goldenberg, and Jane Kang work in the Learning and Teaching Division at Education Development Center (EDC) in Waltham, Massachusetts. EDC is a non-profit organization that designs, implements, and evaluates programs to improve education, health, and economic opportunity worldwide. They are coauthors of the Transition to Algebra program, a research-based classroom resource designed to support student success in algebra by helping them shift their ways of thinking from the concrete procedures of arithmetic to the abstract reasoning that algebra requires. They are also coauthors of the upcoming book, Making Sense of Algebra. Follow EDC on Twitter @EDCtweets.
♦ ♦ ♦ ♦