1

Mathematical Discourse,

Writing, Reading, and

Vocabulary

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Literacy in Math

• What comes to mind?

• Why is it important?

• Who is responsible?

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What Is Literacy in Math?

The ability to use reading and writing, speaking

and listening sufficiently well to engage in

thinking and to communicate ideas. —McKee & Ogle 2005

Students rely on language skills to read, write,

talk, and represent their mathematical thinking

and problem solving. —Fogelberg et al. 2008

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Common Core Standards – Literacy

in the Mathematical Practices

Students . . . understand and use stated assumptions, definitions,

and previously established results in constructing arguments. They

make conjectures and build a logical progression of statements,

They justify their conclusions, communicate them to others, and

respond to the arguments of others . . . making plausible arguments

that take into account the context from which [they] arose.

Students . . . 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 . . . By the time they

reach high school they have learned to examine claims and make

explicit use of definitions.

—From CCSSM Mathematical Practice Standards 3 and 6

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Objectives for the Day

Use reading, writing, and discourse to help students

become more proficient in math.

• Understand levels of discourse and how to develop

and promote meaningful discourse.

• Learn strategies for incorporating meaningful writing

activities that promote learning mathematics.

• Learn about challenges with math vocabulary and

strategies for addressing these challenges.

• Understand the unique challenges with math texts and

math word problems, and learn strategies for

addressing these challenges.

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Agenda

• Introduction

• Discourse

• Writing

• Vocabulary

• Reading

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Whose Job Is It Anyway?

• Why should the math teacher teach

vocabulary, reading, writing, and discussion

strategies?

• Why should the ELA teacher teach math?

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Discourse

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What Would Happen If . . .?

What would happen to the sum below if each

number were increased by 10?

352

+ 49

Will this always be true no matter what two

numbers we add together? Why or why not?

What if it were

a subtraction

problem?

359

– 49

difference

subtract?

10

What Can You Tell Me . . .?

What can you tell me about the value of b

without actually calculating the value of b?

How do you know?

8

10

b

b

Hints:

• Can b < 8? Why or why not?

• Can b > 10? Why or why not?

Will your conclusions about the relationship

between b and a and c always be true?

a b

b c

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Mathematical Discourse

The point of classroom discourse is to

develop students’ understanding of key ideas.

—Adding It Up, 2001

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Higher-Order Questions

Higher-order questions generally challenge the

student to provide additional information and

engage in deeper understanding and

reflection, and ultimately promote greater

conceptual development. —Nathan & Kim 2007

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What Discourse Have You

Experienced Today?

• Think about the math problems you just

worked on and how we processed those

problems through small-group and whole-

group discussions.

• How would you characterize the math talk we

engaged in? Did it further or deepen your

understanding of the math? How?

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Levels of Discourse

Level Type of Question Associated

with This Level of Discourse

Confirm

Recall

Explain

Justify

Generalize

Prove

Is it true?

What is it?

How did you get the answer?

Why is it true?

Is it always true?

What is the evidence that it is true?

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Identify Levels of Discourse

• Read about the levels of discourse.

• Identify the level of discourse for each

student response.

• Share, discuss, and justify with others.

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Discourse Strategies

• Cue

• Wait time

• Rehearse

• Restate

• Revoice

• Add on

Ask, “Why does this make sense?”

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Initiate, Manage, & Conclude

Discourse

• Read over the Strategies to Initiate and

Manage Discourse

• TWPS

– What is one takeaway from this list of strategies?

– What is one question you have regarding one of

these strategies?

– How can you use these ideas in your teaching?

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Planning for Mathematical Discourse

1. Introduce the problem or prompt to the whole class.

(3–5 minutes)

2. Students begin working on the problem individually.

(3–5 minutes)

3. Students collaborate in small groups or with

partners. (5–15 minutes)

4. Whole-group discussion.

(5–20 minutes, or longer if there is time)

5. Whole-group instruction. (same day or next day)

—Adapted from Integrating Literacy and Math, 2010

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Always, Sometimes, or Never

• When is the following statement always true, when is

it sometimes true, and when is it never true?

Subtraction always results in a smaller value.

In other words, If a – b = c, then c < a.

• When is the following statement always true, never

true, and sometimes true?

If a + b + c = d, then d is a multiple of 5.

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What Does 00 Equal? Why?

Which of the following is (are) true and why?

A. 00 = 0

B. 00 = 1

C. 00 is undefined

D. 00 does not exist

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Guess My Polygon

1. Play the game.

2. Switch roles and play again.

3. Create teams of two and play against another team.

How to play:

• Work with a partner. Use the polygon chart.

• Player 1 will pick a polygon from the chart but not tell

player 2 which polygon he/she chooses.

• Player 2 asks yes/no questions to figure out which

polygon player 1 chose. The goal is to be able to

know within three questions.

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Dissecting the Discourse

• Is all discourse time consuming?

• How much time was needed for each part of

the problem?

• What options would you have as the facilitator

with this problem?

• What levels of discourse were involved in

your work on the problem?

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Writing

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Is It Right to Write in Math?

• How much and what kind of writing should

students do in a math class? Why?

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Types of Writing in Math Class

• Vocabulary building

• Note-taking

• Explaining work (answers and processes)

• Reflecting (such as journaling)

• Analyzing (such as Think–Write–Pair-Share)

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Benefits of Writing in Math Class

• Builds vocabulary

• Provides a source for reference

• Clarifies thinking

• Solidifies understanding

• Facilitates processing and deepens thinking

• Prepares students for discourse and further learning

• Assesses knowledge

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More Benefits of Everybody Writes

1. Improved thinking and understanding

2. Students remember twice as much

3. Every student participates

4. Select effective responses

5. Cold call on students

6. Guide students toward what is most important

—Teach Like a Champion, 2010

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Support for English Learners

• Greater challenges but same needs

• Suggestions:

Be attentive to additional vocabulary needs.

Allow students to write in their first language, then

translate.

Allow students to write less, and not as correctly.

Encourage the use of math symbols and diagrams.

Use sentence frames to scaffold writing.

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Quick Write on PARCC Sample

Which of the following is (are) true?

A. 8 x 9 = 81

B. 54 ÷ 9 = 24 ÷ 6

C. 7 x 5 = 25

D. 8 x 3 = 4 x 6

E. 49 ÷ 7 = 56 ÷ 8

What are three

different ways you

could change B to

make it a true math

statement?

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Writing Is Required

Sample Item from PARCC

Type a fraction different than 3/4 in the boxes that

also represents the fractional part of the farmer's

field that is planted with soybeans (the shaded area

on the diagram represents soybean plants).

Farmer’s Field

Explain why the two fractions above are equal.

= 3

4

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Analyze Student Writing

• Analyze the student writing samples for the

problem “Explain why 5 + -7 = -2.”

• What does the writing tell you about student

thinking and understanding?

‒ Are all ways correct?

‒ How are the methods related?

‒ Is there a best method among the four?

Why or why not?

‒ What would be a good next step for these

students or this class?

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Writing Wrap-Up

• Create a prompt for a student writing activity in

math.

• Share your prompt.

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Math Vocabulary

Key Vocabulary from CCSSM Grade 6 Content Standards:

Ratio, rate, unit rate, unit pricing, constant speed, percent,

percent of, common factor, greatest common factor, integers,

positive numbers, negative numbers, rational number, opposite

value, inequality, absolute value, coordinates, coordinate plane,

quadrants, numerical expression, exponent, variable, algebraic

expression, term, coefficient, evaluate expressions, Order of

Operations, properties of operations, distributive property,

equivalent expressions, equations, inequalities, substitution in an

expression, making an equation or inequality true, constraint,

dependent and independent variables, right triangle, volume,

right rectangular prism, edge, face, vertex, surface area, nets for

finding surface area, variability, measure of center, median,

mean, measure of variation, range, interquartile range, deviation,

dot plot, histogram, box plot

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What Does It Mean to Know a Word?

Students progress through five levels of understanding:

1. Have never seen or heard the word before

2. Have seen or heard the word before, but don’t

know what it means

3. Vaguely know the meaning of the word; can

associate it with a concept or context

4. Know a general meaning; understand the word in

reading

5. Know the word well; can explain it and use it in

writing

—Teaching Reading Sourcebook, 2008

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Why Is Vocabulary Important in Math?

• Research shows that reading comprehension

positively affects achievement in arithmetic and

problem solving.

• Vocabulary instruction should focus on specific words

that are important to what students are learning.

(Marzano 2001)

• Math vocabulary is confusing for many students for a

variety of reasons that impede their ability to

understand what they read and hear.

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Challenges with Math Vocabulary

• Double Meanings: Words mean different things in

mathematical vs. nonmathematical contexts.

• Multiple Terms: More than one word can be used to

describe the same concept.

• Symbol Intensity: Math is full of symbols and graphic

representations that carry as much weight as words.

• Homophones: Many math words sound like different

nonmath words.

• Small Words: Many small words make a big

difference in meaning.

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EL Challenges

• Same challenges as regular learners

• Challenges of learning a new language

• Some words have different corresponding

meanings in the first language

– For example, table in Spanish is mesa, and the

mathematical term in Spanish for a math table is

tabla, which in Spanish is a board.

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Identify Challenging Words

• Brainstorm in your groups a list for each of the

following (worksheet provided in PRG):

– Double meanings

– Multiple terms

– Homophones

– Small words or phrases that confuse

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Tier 1 and Tier 2 Words

• Tier 1 – Basic Words: Students know these words

sufficiently on their own, and most students do not

need instruction on these words. For example, and, a,

the, with, etc.

• Tier 2 – Frequent words central to

comprehension: Good candidates for direct

instruction. For example, book, problem, solve,

compute, determine, etc.

—Adapted from Teaching Reading Sourcebook, 2008

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Tier 3 Words

• Tier 3 – Specialized Words: Words that are limited to

a certain field, such as psychology, chemistry, math,

etc. These words are best learned in context. Most

math vocabulary falls into this tier. For example, linear,

rectangle, point, equation, expression, variable, etc.

Some words that are typically in Tier 1 may be Tier 3 in

math. For example, all, and, an, a, some, each, etc.

Many Tier 1 words occur often in math texts and have

precise meanings that students need to recognize.

—Adapted from Teaching Reading Sourcebook, 2008

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When Vocabulary Should Be Taught

• Some words need to be taught prior to use.

• Some words need to be taught in the contexts

in which they arise.

• Vocabulary should be taught at all levels.

• Use of precise definitions should become

more common in middle and high school.

• Key vocabulary should be reviewed and used.

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Consider the Word Variable

• Consider the word variable, which means

A mathematical entity that can stand for any of the

members of a given set.

—The Penguin Dictionary of Mathematics, 1998

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Meaning Can Develop over Time

• Development of meaning and precision for the

word variable:

‒ Elementary school: an unknown quantity

‒ Middle school: a quantify that varies

‒ High school: any of the members of a given set

Variable: A mathematical entity that can stand for any of the

members of a given set. —The Penguin Dictionary of Mathematics, 1998

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Strategies for Teaching Vocabulary

– Parts of word/speech

– Discussion of words/concepts

– Types of vocabulary charts

– Sorting activities

– Graphic organizers

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Parts of Words

• gon having a certain number of angles

• poly many

• oct eight

• hex six

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Parts of Speech

• Verb: add, subtract, sum, rotate, prove, etc.

• Noun: sum, fraction, intercepts, radical, etc.

• Adjective: rectangular, equivalent, sinusoidal, etc.

• Ending: triangle changed to triangular, etc.

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Discussion Challenges

Challenge understanding of key words

Two-Minute Challenge II:

• Can you have the larger half of something?

Why or why not?

• How would students answer and why?

Two-Minute Challenge I:

• Is a square a rectangle? Is a rectangle a

square? Why or why not?

• How would students answer and why?

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Frayer Model Types

• Charts that help organize understanding and

ideas about a single word:

‒ Frayer Model

‒ Word Think Sheet

‒ VVWA: Verbal and Visual Word Association

• Review the samples for each type.

• Create another sample for each type.

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Sorting Activities

• Sort cards or objects into groups based on

common properties or characteristics.

• Students identify the rationale for how

cards/concepts are sorted.

• Each student keeps his/her own record.

• Two examples:

‒ Cue Cards

‒ Concept Sort Activity

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Graphic Organizers

• There are many types of graphic organizers.

‒ Concept map to show connections between

concepts and/or key vocabulary

‒ Agenda to list the day’s activities

‒ Advanced organizer to preview a unit

• Examples for concept maps:

‒ Functions

‒ Numbers

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Why Use Graphic Organizers?

• A concrete way to process, reflect on, and integrate

information and make categorical thinking visible.

• Using graphic organizers is an excellent method of

helping students to visualize the abstractions of

language. Therefore, they are an effective

instructional strategy for English-language learners

(Gersten and Baker 2001).

—Teaching Reading Sourcebook, 2008

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Graphic Organizer for Fractions

Enter the following vocabulary

terms onto the chart. You may

enter additional terms as well.

– benchmark – quotient

– part-whole – ratio

– number line – measurement

– scale factor

– common denominators

– tape diagram

– area model

– compare to whole

– circle diagram

– common numerators

Uses Models

Compare Fractions

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Vocabulary:

– benchmark – quotient

– part-whole – ratio

– number line – measurement

– scale factor

– common denominators

– tape diagram

– area model

– compare to whole

– circle diagram

– common numerators

Uses Models

Compare Fractions

Completed Graphic Organizer

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Vocabulary Wrap-Up

Think–Pair-Share

1. Individual Think Time:

• Which vocabulary activity can/will you do with

your students? Why and how?

2. Pair-Share: Discuss with a partner your thoughts

on the questions above.

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Reading

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The Math Teacher Teaching Reading

Mathematics teachers don’t need to become reading

specialists in order to help students read mathematics

texts, but they do need to recognize that students need

their help reading in mathematical contexts.

. . . most reading teachers do not teach the reading

skills necessary to successfully read in mathematics

class.

—Literacy Strategies for Improving Mathematics Instruction, 2005

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Reading in Math

• Math textbooks

• Math word problems

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Informational Texts

• Student success or failure in school is closely tied to

their ability to comprehend informational text.

• Just as it is important to integrate informational texts

into language arts instruction, so it is important to

integrate comprehension instruction into content-area

teaching, particularly for adolescents (Sadler 2001;

Alvermann and Eakle 2003; Fisher and Frey 2004).

—Teaching Reading Sourcebook, 2008

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Why Students Should Read Math Texts

• Resource for review and learning

• Differentiated instruction

• Independent learners

• Ability to read technical books

• Defense against the substitute teacher

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Density of Text

Mathematics is the most difficult content area material

to read because there are more concepts per word, per

sentence, and per paragraph than in any other subject;

the mixture of words, numerals, letters, symbols, and

graphics requires the reader to shift from one type of

vocabulary to another.

—Braselton & Decker 1994

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Math Symbol Decoding Challenges

Multilevel challenges with math symbols:

• No phonics clues

• Usually must translate into English

• Must connect the symbol to a concept

• Symbols change meaning

– Numerals used in whole numbers, fractions, and

exponents

– Fraction bar means part-whole, division, ratio, etc.

– 25 means 20 + 5, but 2x means 2 times x

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Help Students Understand the Format

• Inform about format of text

• Inform about format of the graphics

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Text Structures

• Description

• Compare/contrast

• Cause/effect

• Problem/solution

• Time order (sequence)

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Math Text Structures

• Introduction

• Explanation

• Examples

• Practice

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Strategies for Effective Reading – 1

• Tapping prior knowledge (or making connections)

• Predicting

• Questioning

• Visualizing

• Summarizing

• Synthesizing

• Monitoring and repairing understanding

—Adapted from From Reading to Math, 2009

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Strategies for Effective Reading – 2

• Think-alouds

• Guided reading

• Elaborative interrogation

• Turn and talk

• Note-taking

• Graphic organizers

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Reading Strategies – You Try It

1. Think-aloud: Scan over 1-2 pages in your math textbook

and to yourself say what is going through your head.

2. Guided reading: Read the same pages you just

scanned over.

– Turn and talk with a partner about the gist of this section.

What is it about?

3. Note-taking: Read the next page and take notes, listing

the key vocabulary. Create your own example that is

like one of the examples in the pages you read.

4. Elaborative interrogation: Answer questions in the PRG.

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Collaborative Strategic Reading (CSR)

BEFORE READING: Preview

Scan – Brainstorm What You Know – Predict What You

Will Learn

DURING READING:

Click and Clunk – Get the Gist

AFTER READING: Wrap Up

Ask and Answer Questions – Review What You Learned

—Teaching Reading Sourcebook, 2008

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Anticipation/Prediction Guide

Tool used to preview a reading, and to focus and

motivate student attention during reading

1. Create questions based on reading.

2. Students answer questions before reading.

3. Students do the reading.

4. Students revisit and change answers as needed.

5. Students provide evidence from text to support

final answers to the anticipation questions.

—Adapted from Teaching Reading in Mathematics, 2002

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Anticipation Guide for Unit Rates

Directions: In the column labeled Me, write T or F next to each statement

based on if you think the statement is true or false. After reading the text,

under Text write T or F for your final opinion for each statement based on

what you have learned from the reading. Be ready to explain how the text

proves your final answer is correct.

Me Text Anticipation Statements

1. A rate is a ratio.

2. All rates are unit rates.

3. All unit rates are rates.

4. Unit rates can be written as fractions or decimals.

5. Equal rates are like equal fractions.

6. Unit rates are a good way to compare prices.

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Create an Anticipation Guide

• Choose a section from your own textbook.

• Create an Anticipation Guide for this section.

• Share with a shoulder buddy.

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Word Problem Challenges

• The main idea is in the last sentence, not the

first sentence.

• Small words make a big difference.

• Students must distinguish key information

from peripheral information.

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K – N – O – W – S

• What I want to KNOW

• What information I NEED

• What OPERATONS I will use

• Show the answer and the WAY to the answer

• Does the answer make SENSE?

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Word Problem Strategies

• Read the last sentence first.

• Read to understand rather than solve.

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Read to Understand the Word

Problem

• Read the last sentence first.

• Read over the whole problem and apply the

first two parts of K-N-O-W-S:

– K: Write down what you need to know.

– N: Record what information is needed or

necessary from the problem.

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Objectives for the Day (How did we do?)

Use reading, writing, and discourse to help students

become more proficient in math.

• Understand levels of discourse and how to develop

and promote meaningful discourse.

• Learn strategies for incorporating meaningful writing

activities that promote learning mathematics.

• Learn about challenges with math vocabulary and

strategies for addressing these challenges.

• Understand the unique challenges with math texts and

math word problems, and learn strategies for

addressing these challenges.

80

End of Day Reflection:

Literacy Strategies in Math

Which strategies were most meaningful to you

and are you most likely to implement? Why?

• Share your thoughts with a partner.

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Thank you!

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