# Teaching and Learning Mathematical Vocabulary

Question: How does Investigations approach the teaching and learning of mathematical vocabulary? Are there vocabulary lists for each unit? Is there a glossary?

Answer:

"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 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 problems clearly and accurately, using whatever words are comfortable for them. However, as you use mathematical terms frequently, students will become used to hearing them and will begin to use them naturally. Even young children can learn to use mathematical vocabulary accurately when they hear it used correctly and in the context of meaningful activities. (Using Mathematical Vocabulary, Mathematical Thinking at Grade 5, page 14.)

What does this look like in the classroom?

To think about what this might actually look like in the classroom, consider this vignette from a teacher on the CGI-Investigations listserve:

 My students had been stumbling over terms as we worked with the Picturing Polygons unit. We've explored and defined polygons, triangles, and angles, and are moving on to quadrilaterals and other polygons. My students expressed a need for reminders around the room to help them with terminology. So we made a list of 16 terms we'd been using, and the kids worked in partners to create posters with definitions and illustrations... They were very engaged, worked hard to make the posters visible from a distance, and tweaked their definitions for both clarity and brevity. For example, the pairs who were working on triangle and quadrilateral came to me and said they knew they could use polygon as part of the definition because the poster for polygon listed its attributes, and anyone in our class reading the word polygon would know those attributes too! In addition, because it's a pet peeve of mine, I wouldn't let them use the word to define the word. For example, they couldn't define right angle as "an angle that measures 90 degrees." It forced them to focus on synonyms and clear meanings, turned out to be a great review and solved the problem of confusing terms for lots of kids. -- Debbie Sauer, 5th grade teacher, Duluth, MN

In this classroom:

• Students can base their definitions on the many experiences they have already had with shapes, angles, etc. (during the first part of the unit Picturing Polygons).
• Students do the work of generating the definition and drawing pictures/examples that illustrate; they also respond and react to each other's work. (Note that students are also developing ideas about mathematical definitions -- they are working towards clear, concise definitions that don't use the word they are defining, or define words that others are defining.)
• Students are deepening and consolidating their knowledge of those shapes and attributes by discussing, comparing, and refining definitions.
• Students have created resources they can continue to use and refer to and adapt.

The goal is that children gain a working knowledge of math ideas and terms, not that they memorize definitions. The mathematical activities in Investigations provide students with numerous opportunities to hear, gain knowledge of, and use mathematical vocabulary in context.

A final note regarding vocabulary lists and glossaries:

The Investigations units do not include a list of vocabulary words. However, when people are looking to create one, the vocabulary support for second language learners IS a good place to start. (This includes the Preview for the Linguistically Diverse Classroom in the Front Matter of each unit, Vocabulary Support for Second Language Learners towards the back of each unit, and an additional, separate support package.

Note that some of the words and terms included in these sections are less about math and more about other words students will need to know and understand about a particular context in order to make sense of the math.

Megan Murray, TERC
Thanks to Kathy Silman and Beth Perry, TERC, and to the members of the CGI-Investigations Listserve, especially Debbie Sauer
December 2003

This information was reprinted with permission of CESAME, Northeastern Univ., and the Educational Alliance, Brown University.