Question: How does Investigations approach the teaching and learning of mathematical vocabulary? Are there vocabulary lists for each unit? Is there a glossary?
"Students do not learn how to speak mathematics by memorizing definitions, but by hearing these words frequently and having many opportunities to use them in context." (Mathematical Thinking at Grade 3, p. 21.)
Teaching and learning math vocabulary can be particularly tricky. The big question is; What does it mean to "know" a vocabulary word or term? For example, say square is a vocabulary word that students must know in Grade X. Does this mean that they can:
- identify or recognize a square?
- draw or construct a square?
- describe a square?
- categorize shapes into squares and not squares?
- explain what makes a square a square (four equal sides, four 90 degree angles)?
- explain why a square is also a rectangle but a rectangle is not a square?
In the early grades of Investigations, students' work with the term square focuses on recognizing/finding/identifying squares, and constructing squares on paper or with materials. There is also a lot of discussion that encourages students to describe squares and to compare them to other shapes such as rectangles, other quadrilaterals, triangles, and other shapes. In the later elementary grades, the focus is on the attributes of shapes and on comparing and categorizing shapes -- what makes a square a square? If all squares are quadrilaterals, are all quadrilaterals squares? If all squares are rectangles, are all rectangles squares?
Square is just one example of a vocabulary word that appears in many grade levels. Although it is a single word, it encompasses several ideas and relationships. The curriculum expects different things of first graders than of fourth graders. What it means to "know" a term depends on the age of the student, and their previous experience.
Whatever the word or term, the attempt in Investigations is to give kids lots of experiences to bump up against an idea, and to build in opportunities to 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 ca