Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently…. In the elementary grades, students give carefully formulated explanations to each other…. —CCSS
The title is potentially misleading. While this standard does include “calculate accurately and efficiently,” its primary focus is precision of communication, in speech, in written symbols, and in specifying the nature and units of quantities in numerical answers and in graphs and diagrams.
The mention of definitions can also be misleading. Elementary school children (and, to a lesser extent, even adults) almost never learn new words effectively from definitions. Virtually all of their 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 their working definitions and make them more precise. For example, the toddler’s first use of “doggie” may refer to all furry things, and only later be applied to a narrower category. In mathematics, too, children can work with ideas without having started with a precise definition. With experience, the concepts will become more precise, and the vocabulary with which we name the concepts will, accordingly, carry more precise meanings. Formal definitions generally come last. Children’s use of language varies with development, but typically does not adhere to “clear definition” as much as to holistic images. That is one reason why children who can state that a triangle is a closed figure made up of three straight sides may still choose as a better example of a triangle than because it conforms more closely to their mental image of triangles, despite its failure to meet the definition they gave.
Curriculum and teaching must be meticulous in the use of mathematical vocabulary and symbols. For example, when students first see the = sign, it may be used in equations like 5 = 3 + 2, or in contexts like 9 + ____ = 8 + 2, in each case making clear that it signals the equality of expressions, and is not merely heralding the arrival of an answer. Teacher Guide information about vocabulary must be clear and correct, and must help teachers understand the role of vocabulary in clear communication: sometimes fancy words distinguish meanings that common vocabulary does not, and in those cases, they aid precision; but there are also times when fancy words camouflage the meaning. Therefore, while teachers and curriculum should never be sloppy in communication, we should choose our level of precision strategically. The goal of precision in communication is clarity of communication.
Communication is hard; precise and clear communication takes years to develop and often eludes even highly educated adults. With elementary school children, it is generally less reasonable to expect them to “state the meaning of the symbols they choose” in any formal way than to expect them to demonstrate their understanding of appropriate terms through unambiguous and correct use. If the teacher and curriculum serve as the “native speakers” of Clear Mathematics, young students, who are the best language learners around, can learn the language from them.