42510011 0010 1010 1101 0001 0100 1011

Building Math Vocabulary Skills

Patty Norman

Logan Toone

Davis School District

4251

0011 0010 1010 1101 0001 0100 1011

Why Build Vocab. Skills?

4251

0011 0010 1010 1101 0001 0100 1011

Why Build Vocab. Skills?

4251

0011 0010 1010 1101 0001 0100 1011

Why Build Vocab. Skills?

4251

0011 0010 1010 1101 0001 0100 1011

Mathematics as a Second Language

The degree to which a student knows a word may vary, and that level of word knowledge has implications for vocabulary instruction. Dale and O‘Rourke (1986) identified four levels of word knowledge:

1. I never saw it before2. I’ve heard of it, but I don’t know what it means3. I recognize it in context - it has something to do with…4. I know it

4251

0011 0010 1010 1101 0001 0100 1011

Mathematics as a Second Language

A particular solution to a differential equation is a function y=f(x) that satisfies the differential equation. A general solution is an expression with arbitrary constants that represents the family of all particular solutions.

What vocabulary has to be known in order to translate the definition?

Defining or translating does not in itself create an

understanding of the language.

4251

0011 0010 1010 1101 0001 0100 1011

ELL Strategies

• Students do not learn mathematical vocabulary by memorizing definitions. Rather, they construct meaning for mathematical vocabulary by actually doing authentic and meaningful mathematics.

• The teacher must be purposeful in constructing learning experiences that direct the student’s attention to specific vocabulary.

4251

0011 0010 1010 1101 0001 0100 1011

• Definitions alone are not sufficient to know a word and it is not necessary to be able to define a word in order to know it. Definitions provide information about the word; however, students also need to know how the word functions in different contexts. For students to learn the word, they benefit from a meaningful explanation of the word, rather than simply a definition. Stahl & Fairbanks (1986) found that providing students with both definitional and contextual information significantly improves comprehension.

4251

0011 0010 1010 1101 0001 0100 1011

4251

0011 0010 1010 1101 0001 0100 1011

ELL Strategies

• Language Objectives– Focus on developing students’ vocabulary

• Brainstorm• Outline • Draft • Revise • Edit• Justify opinions• Negotiate meaning• Provide detailed explanations• Stating conclusions• Summarizing information• Making comparisons

4251

0011 0010 1010 1101 0001 0100 1011

4251

0011 0010 1010 1101 0001 0100 1011

Strategies: Semantic Maps

New Concept

Characteristics

Characteristics

Characteristics

Characteristics

Category

Example Example Example

What it’s Not

4251

0011 0010 1010 1101 0001 0100 1011

Strategies: Semantic Maps

PolynomialFunction

A function consisting of a monomial or a sum of monomials

Terms have coefficients and a variable raised to a power (degree)

In standard form if terms are written in descending order of exponents (L-R)

Functions

f (x) = 3 x 2 + 2 x + 1

• Variables don’t have negative exponents.

• Variables are not in the exponent position.

• Variables can’t have rational exponents (roots, etc.)

f (x) = an x n + an-1 x n-1 + … + a1x + a0

34( )

3f r r5 4 2( ) 9 2 .25 6f x x x x

4251

0011 0010 1010 1101 0001 0100 1011

Try One …

Vertical Angle Pairs

Characteristics

Characteristics

Characteristics

Characteristics

Category

Example Example Example

What it’s Not

4251

0011 0010 1010 1101 0001 0100 1011

Angle Aerobics

Line Ray Line segment Midpoint Acute Obtuse Right Slide/Translation Flip Reflection Turn/Rotation Congruent Similar

4251

0011 0010 1010 1101 0001 0100 1011

Writing

• Students use targeted vocabulary to reflect on and organize their thoughts around related mathematical ideas—e.g. journals, justification of solution strategies, etc.

4251

0011 0010 1010 1101 0001 0100 1011

4251

0011 0010 1010 1101 0001 0100 1011

Quick Pic Book

Vocabulary Self-Selection Strategy:

Personal Dictionaries

“When students are shown how to identify key content vocabulary, they become adept at selecting and learning words they need to know, and given opportunities to practice, comprehension improves.”

(Shearer, Ruddell, & Vogt, 2001)

4251

0011 0010 1010 1101 0001 0100 1011

Journal

4251

0011 0010 1010 1101 0001 0100 1011

4251

0011 0010 1010 1101 0001 0100 1011

Students learn mathematics best by using it, and understanding the language of math gives students the skills they need to think about, talk about, and assimilate new math concepts as they are introduced.

4251

0011 0010 1010 1101 0001 0100 1011

Much instruction focuses students on “doing” mathematics, yet this deprives them of the words they need to communicate what they are learning in relevant ways.

Mathematical language is used and understood around the world, and conventional mathematics vocabulary gives learners the means of communicating those concepts universally.

4251

0011 0010 1010 1101 0001 0100 1011

Using Correct Vocabulary

• An example ….

4251

0011 0010 1010 1101 0001 0100 1011

Definitions of “variable”

--Nichols & Schwartz, Mathematics Dictionary and Handbook

4251

0011 0010 1010 1101 0001 0100 1011

Definitions of “variable”

--Kaplan, McMullin, Algebra to Go: A Mathematics Handbook

4251

0011 0010 1010 1101 0001 0100 1011

Definitions of “variable”

--Holt, Pre-Algebra, 2008

4251

0011 0010 1010 1101 0001 0100 1011

Are we using the proper terms?

4251

0011 0010 1010 1101 0001 0100 1011

When we use “letters”

• All letters are not variables! A letter can represent a quantity that varies (variable), but it can also represent an unknown constant.

4251

0011 0010 1010 1101 0001 0100 1011

What is x ?

• A “variable” or an “unknown constant”?

3 7x

4251

0011 0010 1010 1101 0001 0100 1011

What is x ?

• A “variable” or an “unknown constant”?

3 7 22x

4251

0011 0010 1010 1101 0001 0100 1011

What is x ?

• A “variable” or an “unknown constant”?

3 7y x

4251

0011 0010 1010 1101 0001 0100 1011

What is x ?

• A “variable” or an “unknown constant”?

( ) 3 7f x x

4251

0011 0010 1010 1101 0001 0100 1011

What is x ?

• A “variable” or an “unknown constant”?

3 7 22x

4251

0011 0010 1010 1101 0001 0100 1011

What is x ?

• A “variable” or an “unknown constant”?

1( )

5f x

x

4251

0011 0010 1010 1101 0001 0100 1011

So …

• Should we really be saying – “solve for the variable”– “isolate the variable”– “get the variable all by itself”

4251

0011 0010 1010 1101 0001 0100 1011

Textbooks Even do it Sometimes

-- Houghton Mifflin, 6th Grade Math, 2007

4251

0011 0010 1010 1101 0001 0100 1011

Textbooks Even do it Sometimes

-- McDougall Littell, Algebra 2, 2007

4251

0011 0010 1010 1101 0001 0100 1011

Textbooks Even do it Sometimes

-- McDougall Littell, Algebra 2, 2007

4251

0011 0010 1010 1101 0001 0100 1011

Who Cares?

• Discuss why it might be important to distinguish between constants and variables with your students.