CM MATH SUPPORT PAGES

A F o c u s e d A p p r o a c h

Explicit Language for Content Instruction

TABLE OF CONTENTS

I. Tab One Resources – July 2013 edition

Language Functions (Cognitive Tasks) 5.30 (1.10)

Target Language: Content-specific vocabulary (bricks) 5.31 (1.12)

Target Language: Functional vocabulary (mortar) 5.32 (1.13)

Building Language with Topic “Bricks” & Functional “Mortar” 5.33 (1.15)

Functional Mortar Applied to Different Topics 5.34 (1.16)

Problem Analysis for Language Instruction 5.35 (1.20)

Sample Goal Statements 5.36 (1.27)

II. Tab Two Resources – July 2013 edition

Dominant Functions: Five Sample Passages 5.37 (2.16)

CM Functions – Math Examples 5.38 (2.18-2.27)

Question and Conjecture Function Tool 5.39 (new)

Planning for Mathematical Thinking and Communication 5.41 (2.34)

 

Explicit Language & Literacy for Content Instruction 5.30 (1.10) 2013/E.L.Achieve

Language Functions Language functions identify the tasks – or purposes – of language. They express the cognitive work described in content standards. In mathematics, mathematically proficient students will need to use language to communicate problem solving, reasoning and proof, and representation and connections (CCSS, 2013). Language functions connect how we think to how we express our thinking. Language functions are used to express the cognition described in the Standards for Mathematical Practice.

Identifying the language function in a text helps us understand the author’s purpose. To explain the function of an exponent, a mathematician might use the language of description; to convince the reader that one problem-solving approach is better than another, a writer uses the language of proposition and support; to teach the ‘order of operations’ – the language of sequence and cause and effect. Language functions rely on the use of particular grammatical forms and syntactical structures. For each function, there are distinct language patterns – words and phrases – commonly used to communicate that way of thinking.

Having command of functional language helps students comprehend what they read and convey their understanding, orally and in writing. The language function – the purpose for communicating – determines what linguistic patterns and sentence structures to use. For example, in a series of lessons with the ultimate goal of writing a math problem-solving work sample on using proportions, students first have to comprehend the relationship between the data given in the problem, as well discover the missing piece of data. They have to be able to discuss their reasoning for setting up the proportion, describing which sets of parallel data belong in the numerator and denominator, and explaining the sequence of steps used to cross multiply and divide to find the answer. Finally, they need to check their answer and reflect on their learning.

Language functions move across a continuum from simple (state a preference) to complex (prove a hypothesis). Language functions are used in both oral and written speech. The situation and the content of a particular task determine what language function is needed. Many language functions have both everyday and academic applications (posing a question), while some, such as writing math problem-solving work sample, are specific to an academic context.

In Constructing Meaning, we focus our attention on the five functions that encompass the bulk of the cognitive tasks found in secondary-content course work:

Cause and Effect Compare and Contrast Description and Elaboration

Proposition and Support (argument) Sequence

Common classroom assignments call upon students to utilize the five functions: Comparison Description/Elaboration

Compare characteristics of geometric shapes Deconstruct differing approaches or theories Evaluate opposing viewpoints

Note the characteristics of a line (+/- slope, etc.) Explain the meaning of data in a graph Provide concrete examples of an abstract idea

Cause and Effect/Sequence Proposition and Support Explain one method for problem-solving Articulate the order of events in word problems Use theorems & postulates to complete a proof

Justify an outcome Reflect on meaning Debate the merits of one approach against another

When teachers know from the outset that students must have command of the functional language for retelling their sequence of steps to solve a problem, describing data in a graph, and comparing and contrasting, they can develop lessons that support students in learning and using the specific linguistic structures and vocabulary.

Secondary Constructing Meaning 5.31 (1.12) 2013/E.L.Acheive

TARGET LANGUAGE

Once the dominant language function of a production task is determined, the next step is to identify the target language needed to complete the task successfully. Target language includes not only discipline-specific vocabulary (slope, numerator, exponent), but also the functional words, phrases, and constructions that link the content vocabulary and express the relationships among ideas. These are the linguistic tools of academic discourse, analytical reading and writing, and cognitive rigor.

To support students in acquiring the target language, teachers need guidance regarding what language to teach. The construction metaphor of “bricks” and “mortar” has been helpful in making these decisions. The bricks refer to the vocabulary specific to the content topic at hand; it is what we are thinking, talking, reading, and writing about. Mortar is the functional language that allows us to do something with the bricks; it is how we are organizing our thinking, speaking, and writing about the topic.

The bricks contain the knowledge and information we’re asking students to learn. The mortar makes it possible to apply the knowledge and act on that learning.

Content-specific vocabulary (bricks) Word-level knowledge is critical to comprehension and effective writing. As students develop English proficiency, the vocabulary they learn and practice becomes increasingly sophisticated, growing from words needed to discuss here-and-now, concrete and observable experiences at early stages, to past and future experiences at more intermediate levels, to events not in students’ experience and abstractions at the advanced levels.

This progression moves from:

Basic words (number, add, shape, line, distance) to More specific words and phrases (sum, average, rectangle, slope, parallel) to Increasingly precise terms (median, ordinal number, properties, rhombus, postulate)

Some brick words are common, general utility and are used across multiple topics (subtract, square, time, equal, similar, graph), while others are low-utility words quite specific to a content topic, such as absolute value, parabola, and theorem.

When learning about algebra, topic-specific brick vocabulary will include improper fraction, probability, and x-axis. Learning about calculus calls for knowledge of differential equations, derivative, and integral, while studying geometry calls for knowledge of perpendicular, parallelogram, obtuse angle, etc. So we may teach:

Rectangle Rhombus

Brick words can be defined. They are often the labels in a diagram or the proper names in a timeline. Publishers italicize the bricks or print them in boldface font. Students learn the meaning of bricks through illustrations, graphics, and examples. When most content teachers think about vocabulary, they think of the brick. Bricks are a staple of content teaching.

• All right angles • Parallel sides are congruent • Adjacent sides are perpendicular • Diagonals are not perpendicular

§ Obtuse, acute, or right angles § All four sides are congruent § Diagonals are perpendicular § Diagonals bisect vertex angles

§ Four sides § Parallelograms § Sum of the interior angles

is 360 degrees

Explicit Language & Literacy for Content Instruction 5.32 (1.13) 2013/E.L.Achieve

Functional vocabulary (mortar) To continue with the construction metaphor, “mortar” is the functional language that is required to generate cohesive print and speech. It is grammar in action. It is what keeps us from speaking in single words and phrases.

Functional mortar words and phrases are the parts of the sentence that organize the bricks to express an intended meaning. Without the mortar, we have a list of vocabulary words (or a pile of bricks). Once we decide what we want to communicate, we use our knowledge of grammar and syntax to construct sentences and paragraphs that convey our meaning. We might use the above vocabulary to express understanding with:

• The rhombus and the rectangle have many important similarities. They are both parallelograms and the sum of their interior angles is 360 degrees.

• However, while they both have two pairs of parallel sides, all four sides of a rhombus are congruent, whereas only opposite sides of a rectangle are congruent.

The ability to put grammatical knowledge to use allows us to generate language. The “mortar” is the structure – or frame – of the sentence. It is the language that holds topic-specific words or “bricks” together in a way that enables us to express our thoughts on that topic.

We use language features and structures of English to create the organizational patterns for discourse, reading and writing, complex language, and cognitive processes. Highly proficient English speakers are able to utilize them with agility. These tools include word knowledge and language features, subject/verb agreement, conjunctions to make complex and compound sentences, and word order (syntax).

Mastery of language and syntactical features allows students’ full participation in academics by enabling them to put ideas together in a wide range of ways.

This approach to language development draws on Long’s “Focus on Form” (1988), and does not practice isolated language features. Rather, it focuses on form within a meaning-based context (Doughty and Williams, 1998) and using communicative functions (e.g., using the past tense to describe what happened in a science experiment or historical event) relevant to academic purposes.

Native English speakers automatically use an extensive range of language structures. Proficient English speakers rely on this broad scope of language knowledge to communicate.

English learners must accelerate their learning of how English works. This includes learning to use not only a multitude of words, but also a breadth of language patterns to communicate relationships between and among ideas. The intentional and purposeful teaching of language structures – the mortar – enables English learners to internalize the patterns needed to express concepts, ideas, and thinking.

As complexity of academic content increases, the learner must be able to know and use increasingly complex language structures. Consider these examples of sequence:

First, I subtract 3 from both sides of the equation. Next, I divide both sides by the coefficient 7. To solve the equation, I must isolate the variable x. The first step is to subtract 3 from both sides.

Then I can solve for x by dividing both sides by the coefficient 7. There are two steps needed to solve this equation. The initial step is to subtract 3 from both sides.

I can then isolate the variable x by dividing by sides by the coefficient 7.

Secondary Constructing Meaning 5.33 (1.15) 2013/E.L.Acheive

Building Language with Topic “Bricks” & Functional “Mortar” To illustrate the importance of addressing both brick and mortar vocabulary in content instruction, let’s look at a few examples of the relationship between function and topic:

Function

Determines “Mortar”

Topic

Determines “Bricks”

Sample Sentences

Functional language (mortar) is bolded.

Cause & Effect

Interpreting Graphs and

Charts

As a result of lower ticket sales, the total revenue decreased.

Due to the rise in demand for the product, the price increased.

If you roll the die 100 times, then you will get about the same number of even and odd results.

Compare & Contrast

Geometry Although obtuse angles can be found in both pentagons and octagons, the sum of the interior angles in these two polygons is completely different.

All of the quadrilaterals we’ve studied have the same sum of their interior angles.

Like a rhombus, rectangles are quadrilaterals. However, the diagonals of a rhombus are perpendicular to each other and those of a rectangle are not.

Description/Elaboration

Rational Numbers

An improper fraction is when the numerator is greater than or equal to the denominator.

A simple fraction is an example of a rational number.

Square roots are often associated with irrational numbers.

All rational numbers can be described as a ratio of two integers.

Proposition/ Support

(Problem/ Solution)

Solving Linear

Equations

I learned that solving linear equations with the substitution method leads to the same result as using the graphing method to see where the lines intersect.

It is clear that the substitution method is more precise than trying to determine the exact point of intersection of two lines on a grid.

I discovered that the elimination method of solving linear equations is a good option when one variable from both equations has the same coefficient.

Sequence Order of Operations

Simplifying algebraic expressions begins with doing operations in parenthesis first.

The process continues with simplifying the terms with exponents.

Later, I can multiply and divide from left to right.

By the end, addition and subtraction from left to right are the only remaining operations.

Explicit Language & Literacy for Content Instruction 5.34 (1.16) 2013/E.L.Achieve

Below are examples of how functional mortar can be applied to different topics:

Cause & Effect

Because the slope-intercept form of the equation was used to graph the solution, there was no need to create a table.

The dice rolling probability experiment led to an outcome that matched the prediction from the probability formula.

Compare & Contrast

While ∆ABC and ∆XYZ are both right triangles, there are several major differences between them. The most notable is that ∆ABC is an isosceles triangle and ∆XYZ is a scalene triangle.

Description/ Elaboration

A parabola consists of a curve with each point equidistant from the focus and the directrix. Another attribute is that it is symmetrical across its axis.

Proposition/ Support

The fact that point A satisfies both equations proves that it is the point of intersection of the two lines.

It is easy to see that this is the correct answer, because when we evaluate each side of the equation, the equality holds true (i.e. 0=0).

Sequencing

Initially, we find the greatest common factor (GCF) of the numerator and denominator. Subsequently, we simplify the fraction by dividing the top and bottom by that factor.

To begin, add all of the test scores in the class. After that, divide the sum by the number of tests to find the mean.

Utilizing Our Expertise (Think, Write, Pair Share)

Directions: Think of an essential topic from your content area. Use the samples above and on the previous page as a guide to write five sentences, one for each of the listed functions.

Topic from your content area:

Cause and Effect:

Compare and Contrast:

Description and Elaboration:

Proposition and Support:

Sequence:

Constructing Meaning for Mathematics

5.35 (1.20) Dutro & Levy © 2013/E.L.Achieve

Problem Analys is for Language Instruct ion

Assessing Performance - sample student response

What will students need to say, write or do to explain their thinking?

Language Production - functions

Which language function(s) will students be expected to produce?

Making Sense of the Task Description/Elaboration

Question / Conjecture

Communication Reason Sequencing

Cause and Effect

Reflecting and Evaluation Compare and Contrast

Proposition / Support

Language Production - bricks

What content-specific language will students be expected to know and use?

Language Production - mortar

What functional words and phrases will students be expected to produce?

Student Learning Goal

Students will understand _________________________________ and

(content goal)

be able to _____________________________________________.

(language goal)

Explicit Language & Literacy for Content Instruction 5.36 (1.27) 2013/E.L.Achieve

Linear Equations

Students will understand the methods of solving systems of linear equations using substitution or elimination and be able to compare and contrast their similarities and differences in a ticket out the door summary.

Students will understand the term “slope-intercept form” and be able to explain key characteristics of each element of this linear equation in a short constructed response.

Students will understand how to calculate the slope of a line perpendicular to a given linear equation and use cause and effect language to describe this process verbally to a partner.

Students will understand how to write the equation of a line given two points and be able to write the sequence of steps needed for this operation using full sentences in a dialectical journal.

Probability

Students will understand how probability works with dice and discuss the impact it has on strategy for the game of Yahtzee in a small group conversation.

Students will identify the components of a probability experiment and be able to explain their purpose in discovering the likelihood of any particular event occurring in a verbal dialogue with a partner.

Students will identify two possible outcomes of a probability experiment and write a summary comparing the likelihood of each event.

Students will understand the term “probability line” and be able to sequence different types of events from least likely to most likely to occur using complete sentences for each event.

Geometry

Students will understand the term polyhedron and be able to describe the characteristics as well as the types of these solids in a ticket out the door.

Students will compare the characteristics of equilateral, isosceles, and scalene triangles and explain their thinking in a paragraph.

Students will be able to define the term golden ratio and describe its characteristics as well as its application to geometry, art, and architecture in a 5 paragraph essay.

Students will understand and be able to summarize the steps that led to finding the surface area of a pyramid using full sentences in a dialectical journal.

PRACTICE AND APPLICATION Directions: Review the sample goals above from your subject area. Working with a partner, identify the content objective, language objective, and product for each goal. Then, try writing a learning goal of your own.

5.37 (2.16) Explicit Language for Content Instruction Dutro & Levy 2013/E.L.Achieve

DOMINANT FUNCTIONS: FIVE SAMPLE PASSAGES The model paragraphs below demonstrate how each of the five dominant functions can contribute to the solution, discussion, and evaluation of a math problem.

Given the following ordered pairs and corresponding line segments, determine the geometric shape. Use everything you have learned about the characteristics of shapes. Provide an explanation of your reasoning.

A (0,0), B (4,3), C (9,3), D (5,0); 𝐴𝐵, 𝐵𝐶, 𝐶𝐷, 𝐴𝐷

Sequence

Initially, I plotted the four points on graph paper. Then I connected the points for the line segments listed. Next, I looked at the dimensions and characteristics of the shape and made a prediction. After that, I calculated the slope of each line segment and determined mathematically that the opposite sides are parallel. Subsequently, I calculated the length of each line segment and found that they are congruent. Ultimately, I matched the traits of the shape with the attributes of common quadrilaterals and discovered that it is a rhombus.

Compare & Contrast

To determine the shape, I compared the characteristics of this shape to those of two common quadrilaterals: the rhombus and the rectangle. The rhombus and the rectangle have many important similarities. They are both parallelograms and the sum of their interior angles is 360 degrees. However, while they both have two pair of parallel sides, all four sides of a rhombus are congruent, whereas only opposite sides of a rectangle are congruent. Another distinction between the two shapes is that all of the interior angles of a rectangle must be right angles, unlike rhombi, which can have obtuse, acute, or right angles. A rhombus with right angles is a square. Finally, diagonals of rhombi are perpendicular, in contrast to those of rectangles, which are not.

Description and Elaboration

A quadrilateral is defined as a shape with four sides. This shape is a specific type of quadrilateral. Opposite sides 𝐴𝐵 and 𝐶𝐷 are parallel, each having a slope of ¾. Opposite sides 𝐵𝐶 and 𝐴𝐷 are also parallel with a slope of zero. Parallelograms belong to the family of quadrilaterals. They are four-sided shapes characterized by two pairs of parallel sides. This shape is consistent with the traits of a parallelogram. A rhombus can be further identified as a subset of parallelogram with the additional attribute that all sides have equal length. Each side of this shape is five units long. This shape is an example of a rhombus.

Cause & Effect

I calculated the slope of the line segments in order to determine if the opposite sides of the shape are parallel. Due to the fact that one pair of opposite sides has a slope of ¾ and the other pair has a slope of zero, I was able to determine that the shape is a parallelogram. Since all four sides are congruent, I was able to narrow the options down to a square or a non-square rhombus. The interior angles are not right angles. Therefore, the shape is not a square, but still a rhombus.

Proposition & Support (Problem/Solution)

I believe this shape is a rhombus. The evidence is clear that this shape matches the characteristics of a non-square rhombus. It has four sides and the opposite sides are parallel. Most importantly, I can distinguish it from a rectangle based on the fact that all four sides are congruent. My partner predicted that it would be a square. I disagree with this position for the following reason: it has no right angles. According to my list of traits of geometric shapes, a square is a subset of rhombus with all right angles. This shape has two obtuse and two acute angles. The attributes of this shape support my claim that it is a rhombus.

Dutro & Levy Constructing Meaning: Explicit Language for Content Instruction 5.38 (2.18-2.27) 2013/E.L.Achieve

CM FUNCTIONS – MATH EXAMPLES

Elaboration/Description

Compare & Contrast

Cause & Effect

need roll of 1 or 5 2 positive outcomes

2 in the numerator 6 in the denominator

ratio of 2/6 simplifies to 1/3

6 sides on a die 6 total outcomes Due to the fact that there are six sides on a die, the probability of rolling a 1 or a 5 with one die to continue a round of the game Farkle is 1/3. This is because the numbers 1 and 5 represent two possible positive outcomes out of six total possible outcomes. Therefore, the two possible positive outcomes becomes the numerator and the six total possible outcomes becomes the denominator. This leads to a ratio of 2/6 which in turn can be simplified to 1/3.

Proposition/Support

Proposition: 2 discounts with a sum larger than a single discount is not always better. Evidence: 50% off > 15% off on top of 40% off. Result: Always calculate each option.

Sequence

x2 + 8x + 16 = 0 Factor

(x )(x )

(x + 4)(x + 4)

x + 4 = 0

x = -4

Rhombus Category Rectangle 2 pairs of parallel sides

parallel sides

2 pairs of parallel sides

all sides congruent

congruent sides

opposite sides congruent

acute, right, or obtuse

angles only right angles

SLOPE-INTERCEPT

FORM

Isolate y on left side

m is the slope

x is on the right side

b is the y-intercept

The rhombus and the rectangle have many important similarities. However, while they both have two pairs of parallel sides, all four sides of a rhombus are congruent, whereas only opposite sides of a rectangle are congruent. Another distinction between the two shapes is that all of the interior angles of a rectangle must be right angles, unlike rhombi, which can have obtuse, acute, or right angles.

It is clear that two discounts with a sum larger than a single discount do not always produce the best deal. The evidence for my position is that a 50% discount is better than a 15% coupon on top of a 40% off sale. This result suggests that shoppers should always calculate each option before buying.

To begin solving the quadratic equation x2 + 8x + 16 = 0, I had to factor the expression on the left into two binomials. Next, I noted that the first term, x2, has a coefficient of 1, so I knew that the x in each factor would have a coefficient of 1. Subsequently, I found two numbers with a product of +16 and a sum of +8. Following those steps, I was able to factor out (x+4)(x+4). Finally, I set each factor equal to zero and solved for x. The solution is x equals -4.

The slope-intercept form of a line refers to the equation of the line that isolates ‘y’ on one side. The other side consists of the ‘x’ variable with a coefficient, represented by an ‘m’ in the generic form, that is the slope of the line. This side includes the constant, ‘b’, which is the y-intercept.

5.39 (new) Constructing Meaning: Explicit Language for Content Instruction Dutro & Levy 2013/E.L.Achieve

QUESTION AND CONJECTURE Helpful Signal Words Sample Sentence Frames

Inte

rmed

iate

what

who

when

why

how

guess

assume

imagine

predict

claim

What do you mean by ____________________________?

How did you come up with ________________________?

Until we know ___________, we can’t assume ________.

After reading __________ and examining ___________, I predict ________________________________________.

Ea

rly

Ad

van

ced

question

examine

doubt

hypothesis/theory

wonder

estimate

infer

suppose

suspect

anticipate

My hypothesis is that _________ affects _____________.

Given the relationship between _________ and ________, what can we infer about __________________________?

_____________ makes me wonder if ________________.

By estimating ___________, I can check _____________.

Ad

van

ced

interrogate

inquire

query

presume

dispute

conjecture

speculate

surmise

postulate

extrapolate

Using what we know about ____________________, what can we speculate/conjecture about __________________?

If we extrapolate the trend in _________, we can postulate that ___________________________________________.

Even though ____________________ is indisputable, the subsequent assumption that ______________ is dubious.

Related Functions & Sample Frames

Gather information

Would you please give an example of __________________________________________?

Why is it that _______________________ when _________________________________?

How could you verify _______________________________________________________?

Predict an answer or result

with partial information

Based on __________________, one could surmise/presume _______________________.

If __________________________, wouldn't that also mean that ____________________?

Without knowing the outcome/solution, my educated guess is that ___________________.

Evaluate the accuracy of an

answer or claim

Did you consider ____________________ when _________________________________?

What criteria/data was used to determine/evaluate ________________________________?

Does the hypothesis/theory/postulate hold true when ______________________________?

What would the result/answer/solution be if _____________________________________?

Dutro & Levy Constructing Meaning: Explicit Language for Content Instruction 5.40 (new) 2013/E.L.Achieve

QUESTION AND CONJECTURE Helpful Graphic Organizers

Geometry Example

Pre-Algebra Example

Target offered a 15% off coupon for jeans that were already on sale for 40% off. Old Navy had a coupon for 50% off the original price of a pair of jeans that were the same price. Which was the better deal?

Known Information

Prediction Clues What was the actual outcome?

Target’s offer is an extra 15% off an item already on sale for 40% off.

Old Navy’s offer is 50% off.

I presume that the double discount at Target is the better deal.

Target’s 40% + 15% seems to be a larger discount than Old Navy’s 50% discount by itself.

When I did the calculation, I discovered that my assumption was wrong. The 50% coupon is the better deal.

My initial hypothesis was incorrect. I assumed that two discounts with a sum larger than a single discount would always produce the best deal. However, I discovered that this theory is not always true.

KNOWN INFORMATION

PREDICTION CLUES FINAL OUTCOME

KNOWN

INFORMATION

Prediction 1

Prediction 2

Prediction 3

Prediction 4

If two congruent circles overlap as in the diagram, will their center points (A and B) form an equilateral triangle with a third point C, where the circles cut one another? Make a prediction and explain your reasoning. In the next lesson you will prove your claim using the postulates studied in class. Prediction: I speculate that the triangle formed by points A, B, and C is an equilateral triangle. My conjecture is based on the fact that the length of line segment AB represents the radius of both circles. Since line segments AC and BC also represent the radii, all three sides of the triangle must have the same length, thus making an equilateral triangle.

Explicit Language for Content Instruction Dutro & Levy

5.41 (2.34) 2013/E.L.Achieve

Planning for Mathematical Thinking and Communication

The functions Cognitive and linguistic

The thinking Modeling and demonstration

The language Brick and mortar

The routines Oral and written

Compare & Contrast Think Alouds Sentence frames Think, (Write), Pair, Share

Cause & Effect Graphic organizers Word banks Talking Chips

Elaboration/Description Manipulatives Frayer model Numbered Heads

Proposition & Support Real world context Writing templates Writing to Learn

Sequencing

Laying the Groundwork - background knowledge about the content and language What explicit instruction is required to support students in making sense of the task?

Learning the Material - skills and language needed to express understanding of the content What explicit instruction is required to support students in communicating their reasoning?

Applying the Knowledge - producing the final outcome

What explicit instruction is required to support students in reflecting and evaluating?