 discuss it. Once students are familiar and comfortable with a concept they often search for, or can much more easily make sense of, the more specific mathematical vocabulary that is associated with it. Consider this quote, from the Preview for the Linguistically Diverse Classroom:
"In the Investigations curriculum, mathematical vocabulary is introduced naturally during the activities. We don't ask students to learn definitions of new terms; rather, they come to understand such words as factor or area or symmetry by hearing them used frequently in discussion as they investigate new concepts. This approach is compatible with current theories of second-language acquisition, which emphasize the use of new vocabulary in meaningful contexts while students are actively involved with objects, pictures, and physical movement." (Page I-22 of Coins, Coupons, and Combinations)
This doesn't mean that teachers shouldn't use accurate mathematical vocabulary, or encourage students to do so when they are ready. Consider these excerpts from Teacher Notes, first from K-2 and then from 3-5:
At this age, students will not use many conventional mathematical terms. For example, they will probably not know that a blue pattern block is a rhombus or parallelogram, but may instead use the everyday term diamond. They are likely to know some geometric names, such as square, circle, and rectangle, but may sometimes apply these incorrectly. For example, they might call a cube a square or call a triangle a rectangle.
Enter into students' conversations, often using the same terms they are using, but also asking them questions or making comments that challenge them to be clearer and more precise. For example:
Claire: I'm using diamonds all around the edge.
Are you going to use all blue diamonds, or are you going to use some of the tan diamonds?
You can also use interactions to introduce conventional mathematical names for two- and three-dimensional shapes, so that students hear these terms used in context:
Luis: Look, three of these blue ones can fit right on top of the yellow one.
Luis noticed that three of these diamonds can fit right on top of the hexagon. Did anyone notice any other pieces that can fit together on the hexagon?
You don't need to insist that students use the conventional terms. They will begin to learn them naturally, the same way they learn other vocabulary -- by hearing them used correctly in context. In geometry activities in later units, the students will have many experiences in classifying, describing, and defining shapes. (Talking about Shapes, Mathematical Thinking at Grade 1, page 11.)
Students learn mathematical words the same way they learn other vocabulary -- by hearing them used correctly, frequently, and in context. Young children learn words by hearing their families use language appropriately. When they make mistakes -- "Look at the big doggie," says a young child pointing at a horse -- family members use the correct word, and the child gradually learns the distinctions among all the four-footed animals that at first look like "doggies."
We don't ask young children to memorize definitions of horse or dog; rather, we find opportunities for them to hear words being used in meaningful ways. Learning mathematical vocabulary is no different.
When you speak to students, try to use mathematical vocabulary as often as you can. Connect mathematical terms with more familiar words that students may know, and use these terms in context that make sense to students. For example:
Antonio and Kevin made a rectangle from 24 tiles. The rectangle is 8 tiles across and 3 tiles down. Its dimensions are 8 by 3.
While we counted around the class by 25's, Toshi listed the numbers we said on the board. Look at what she wrote. Do we have the multiples of 25 on the board?
Throughout the Investigations curriculum, we will point out mathematical terms that are important for you to use in context. Don't insist that the students use these terms. What's important is that they express their ideas and describe their strategies for solving mathematical 