Language Arts Literacy and Math Literacy: An Integrated Perspective

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Language Arts Literacy and Math Literacy An
Integrated Perspective

• Bill Crombie
• bcrombie_at_aol.com

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Basic Definition

• Orality
• Listening Speaking
  Reasoning

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Basic Definition

• Literacy
• Reading Writing
  Reasoning

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Models of Mathematics as Language

• Between Language Model - Pidgin
• Within Language Model - Register

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Between-Language Model of Mathematics ( Pidgin )

• Focus on Key Words (Minimal Translation
  Dictionary )
• Focus on the Rules for Symbols
• Focus on Specific Numerical Values

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How to Solve Word Problems in AlgebraMildred
Johnson and Tim Johnson

• Facts to remember Times as much means
  multiply.More than means add. Decreased by
  means subtract.Increased by means add.
  Percent of means multiply.Is, was, will be
  becomes the equal sign () in algebra.

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Within-Language Model of Mathematics ( Register )

• Focus on Ordinary Language Descriptions
• Focus on the Meaning of Symbols
• Focus on Quantitative Relationships

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Reading and Writing Systems

• Phonographic Symbol SystemsSymbols represent
  sounds.
• Logographic Symbol SystemsSymbols represent
  ideas.
• Diagrammatic Symbol Systems Graphic elements
  represent relationships among ideas.

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Language Arts Literacy and Math Literacy An
Integrated Perspective

• Reading Mathematics

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Spelling-Pronunciation Reading

• 
  
• Context
• 
  interpretation
• Word Meaning
• decoding
• Symbol

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Interpretive Reading

• 
  
• Context
• 
  encoding
• Word Meaning
• 
  
  interpretation
• Symbol

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Formal Reading of Mathematical Symbols

• 
  
• Context
• 
  interpretation
• Word Meaning
• decoding
• Symbol

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Interpretive Reading of Mathematical Symbols

• 
  
• Context
• 
  encoding
• Word Meaning
• 
  
  interpretation
• Symbol

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Fraction Example

• 34
• 3 over 4
• 3 parts out of 4 parts
• 3 compared to 4
• 3 measured by 4
• 3 for every 4
• multiplication by 3, division by 4

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• Concepts
  Symbols Relations
• O
  O
  Unique concept symbol
• 
  O
• O
  O
  One concept
• 
  
  for many symbols
• 
  O
  (synonym)
• O
• O
  O
  Many concepts
• 
  
  for one symbol
• O
  
  (homonym)

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• Mathematicians have a habit, which is puzzling
  to those engaged in tracing out meanings, but is
  very convenient in practice, of using the same
  symbol in different though allied senses. The one
  essential requisite for a symbol in their eyes is
  that, whatever its possible varieties of meaning,
  the formal laws for its use shall always be the
  same.
• An Introduction to Mathematics Alfred North
  Whitehead

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Representations of Information

• Visual
• Verbal
  Algebraic
• Geometric

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Geometry provides the Visualizations

• Direct Observations
• Observation Sentences
• Equations
  Inequalities

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Observation Sentences

• 2( x a )

a
x
x
a
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Observation Sentences

• 2x 2a

a
x
x
a
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Observation Sentences

• The Distributive Property is a description of
  equivalent arrangements of the same quantities.
• 2( x a ) 2x 2a

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Multiple Readings of Diagrams

• a b c
• a b c

b c
a
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Multiple Readings of Diagrams

• a b c
• a b c

a b
c
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Multiple Readings of Symbols

• a b

• a minus b
• a take-away b
• a compared to b

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Multiple Readings of Symbols

• a b

• a plus b
• a put-together-with b
• a followed by b
• a moved by b

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Reading Mathematics

• 4x -5 -3x 16

-3x 16
decoding
-5 4x
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Formal Reading of Mathematical Symbols

• 1x 1 gt -2x 4 1x gt -2x 3
• 3x gt 3x gt 1 x is greater than 1.

1
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Interpretive Reading of Mathematical Symbols

• 1x 3 gt -2x 6 ( 1x 3 ) is above ( -2x 6
  ) x is to the right of a positive number
  x gt a

y2 -2x4
y11x1
a
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Text Structure

• How information is distributed in a text

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The Text-Books Logical Structure

• Definition
• Theorem/Procedure
• Examples

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• To many, mathematics is a collection of
  theorems. For me mathematics is a collection of
  examples a theorem is a statement about a
  collection of examples and the purpose of proving
  theorems is to classify and explain the examples.
  
• Subnormal Operators
• John Conway

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A Problems-to-Principles Sequence

• Examples
• Definition
• Theorem/Procedure

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Language Arts Literacy and Math Literacy An
Integrated Perspective

• Writing Mathematics

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Historical Stages in the Development of
Algebras Writing System

• Rhetorical
• Syncopated
• Symbolic

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• Writing restructures consciousness.
• Orality and Literacy
• Walter Ong

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Writing Mathematics

• 4x -5 -3x 16

-3x 16
encoding
-5 4x
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Problem Types addressed by Writing

• Problems of Description
• Problems of Procedure
• Problems of Explanation Typically the
  primary type of writing in the mathematics class.

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Learning Cycle

• I. Exploration
• II. Reflection
• III. Theory
• IV. Application

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4 Question Reports

• I. What happened?
• II. Why?
• III. What conclusions?
• IV. What implications?

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4 Question Reports

• What happened? ( Narrative writing )
• Why? ( Persuasive writing )
• What conclusions? ( Informative writing )
• What implications ( Persuasive writing )

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Language Arts Literacy and Math Literacy An
Integrated Perspective

• Reasoning in Mathematics

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Nominalization

• How verbs become nouns

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Nominalization ( Verbs to Nouns )

• Actions (Verbs) Fair Sharing Comparing
• Operations Division Subtraction
• Objects (Nouns) Fractions Integers

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Reasoning

• Examples
• Conjectures
• Proof / Counter Examples
• Theorem

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Reasoning

• Evidence Claim
• Justification Qualifications
• The Uses of Argument
• Stephen Toulmin

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Language Arts Literacy and Math Literacy An
Integrated Perspective

• Reasoning with Diagrams

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• By diagrammatic reasoning, I mean reasoning
  which constructs a diagram according to a precept
  expressed in general terms, performs experiments
  upon this diagram, notes their results, assures
  itself that similar experiments upon any diagram
  constructed according to the same precept would
  have the same results, and expresses this in
  general terms.
• The New Elements of Mathematics
• Charles S. Peirce

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Reasoning with Diagrams

• a b is the displacement from b to a.
• The displacement -b moves from point b to the
  origin.
• The displacement a moves from the origin to
  point a.
• a b a -b

0 a
b
-b
a
a a b
-b
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Reasoning about Inequalities

• Regions are typically described by evaluating a
  linear form for specific values of x and y. For
  example, when x 0 and y 0, then 4x 3y 9
  is negative. So the region to the left of the
  line is described as
• 4x 3y 9 lt 0 .

4x 3y 9 0
4x 3y 9 gt 0
4x 3y 9 lt 0
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Reasoning about Inequalities
4x 3y 9 0

• There are four sets of displacement pairs in the
  plane. Two of the pairs define the points on the
  line.
• The other two pair determine which side of the
  line is greater than zero and which side of the
  line is less than zero.

4x 3y 9 gt 0
4x 3y 9 lt 0
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Language Arts Literacy and Math Literacy An
Integrated Perspective

• Reasoning with Symbols

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Reasoning with Symbols

• a b a 0 b
• 0 -b b
• a b a -b b b
• a b a -b

• a b is not changed by adding zero.
• Zero can be written as -b plus b.
• b minus b is zero.
• Subtracting b from a yields the same result as
  adding -b to a.

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• Literacy
• as
• Reading, Writing, and Reasoning
• Language Arts
  Mathematics

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• By the aid of symbolism we can make
  transitions in reasoning almost mechanically by
  the eye, which otherwise would call into play the
  higher facilities of the brain. It is a
  profoundly erroneous truism that we should
  cultivate the habit of thinking of what we are
  doing. The precise opposite is the case.
  Civilization advances by extending the number of
  operations which we can perform without thinking
  about them.
• An Introduction to Mathematics Alfred North
  Whitehead