Mathematical Literacy: Reading and Writing Count in Mathematics

Learning

Julie LearnedUniversity of MichiganJanuary 28, 2010

Learning Goals Define mathematical literacy Design instruction for translating across multiple forms of representation in mathematics

Analyze mathematical texts and identify the reading demands students will encounter

Use a variety of teaching tools to support students’ reading

Design instruction for communicating mathematical understanding in writing

Agenda Mathematical Literacy Multiple Representations in Mathematics

Mathematical Text Analysis Teaching Tools for Reading Comprehension

Communicating Mathematical Understanding in Writing Teaching Tools for Writing

What is mathematical literacy?

Review the following images and answer these questions:

Is this mathematical literacy? Is the person engaged in an act of mathematical literacy?

Reading math textbook

QuickTime™ and a decompressor

are needed to see this picture.

World-famous mathematician, Qiu Chengtong, lecturing

What counts as mathematical literacy? Consider the images of people engaged in mathematical thinking, writing, reading, speaking, and problem solving

Jot down your own definition of mathematical literacy

What’s unique about mathematics? Knowledge is certified by means of a deductive proof.

Mathematicians make claims of absolute certainty.

Words, terms, symbols, and diagrams have precise, shared meanings.

Mathematicians navigate multiple symbol systems. Mathematical knowledge builds on and does not discard what came before. Mathematical research and literature is stable and reliable.

The knowledge system must be free of logical contradiction.

Adapted from mathematician, Dr. Hyman Bass, 2006

Mathematical Literacy Navigate across multiple symbol systems Read and make meaning with mathematical texts

Understand and manipulate mathematical concepts to create mathematical arguments

Communicate mathematical understanding Make sense and evaluations of “real-world” mathematics

Engage in a way of knowing and seeing the world, a way of problem solving that goes beyond just doing the math

Multiple Representations in Mathematics - What’s the connection?

Linear Functions

are particular kinds of correspondences between two

numerical magnitudes X and Y

Graph

that when graphed on a Cartesian coordinate system fall on a Euclidean line y x whose slope (tangent of the angle with the horizontal axis) equals the constant

ratio 1212xxyy−−

Formula

that can be expressed as the set {∈xyx),,( and ,Yy∈such tha t}bxay+⋅= where aequals the constan t ratio

1212xxyy−− and b is the y-value

associated wit hx=0

Problem

that model application problems such as transforming measures due to a change of a measurement scale (e.g., the transformation between temperatures in degrees Fahrenheit and those in degrees Celsius).

Table

that when displayed in tables such as X Y 1x 1y 2x 2y 3x 3y 4x 4y

verify that

12123434 xxyyxxyy −−−−=

for any pair o f associated values yx→

Navigating Multiple RepresentationsExercise 1

Reading Mathematical Texts Read each of the “texts.” List specific knowledge you put to use to understand what you read.

List ways of reading and ways of knowing you use in order to understand.

Navigating Multiple RepresentationsExercise 2

Role Play Role play with a partner. Teach your partner how to make sense of the particular way in which each text represents a linear function.

Include explanations of how you read each text.

Articulate what you did as a reader to make meaning.

Navigating Multiple RepresentationsDebriefing Exercises 1 and 2

X Y=f(x) -9 -9 -3 3 6 21

18 39

Analyzing Mathematical Texts

Complex word problems Examine the nature of the text Predict the relationship between text and reader

Analyze and plan for relationships across texts

Text AnalysisHomework 8: To Kearny by Equation

Answer relevant text analysis questions with a partner

Note Homework 8 is two pages Create a chart with your table that lists what you learned from the analysis (e.g., key text features, reading demands, challenging vocabulary, what students need to know before reading)

Text AnalysisDebriefing Homework 8

Long text with a lot of mathematical information and multiple questions

Lots of extraneous information Vocabulary: emigrants, ferries, Fort Laramie, Papan brothers, profit, expense

Necessary knowledge: variables, substitution, manipulating equations, knowledge of Overland Trail, ability to connect math to “real-life” problems

Multiple representations: words, equations, picture

Now what? Given the reading demands of the text, what do we do about it?

Many ways to support students’ reading

Choose based on what you know about the students, the text, and the context.

Read aloud, underlining, pair and share reading, graphic organizer

Teaching Tool: Reading Mathematical Word Problems What is the word problem asking; what is the main question?

List any words or concepts you do not understand. Get an explanation or definition.

List all of the relevant, important information for this question.

Answer the question using the relevant information. Show all of your thinking. You may use words, numbers, symbols, pictures, graphs, charts, and tables.

Justify the mathematical approach you took to solve this problem. Write down any questions or confusion you still have about the problem.

Reading Mathematical Word Problems Homework 8

Use the graphic organizer to read and answer Question 1 or 2 in Homework 8.

Assume you have a family unit of 14. Assume your group’s four family units have 52 people in total.

When finished, debrief with a partner: in what ways might this support mathematical knowledge building and literacy?

Communicating Mathematical Understanding in Writing Deepens understanding Requires technical vocabulary Requires knowledge about how to structure mathematical arguments

Requires opportunities to practice writing about mathematics

Teaching Tools for Writing

1. Best work reflections2. Frayer model of concept

development3. Building vocabulary: word

walls, word cards, word sort4. Summarizing mathematics texts

with GIST

Best Work Reflection How do you demonstrate your learning in this work?

What academic strengths do you display in this work?

How did you improve your skills and learning by completing this work?

What learning from this work will you remember most at this time next year?

How will you take your strengths from this work and apply them to other projects in or out of school?

Frayer Model of Concept Development

Definition in your own words

Characteristics

Examples Non-examples

Concept

Adapted from Frayer, Federick, & Klausmeier, 1969

Frayer Model of Concept Development Example

Definition in your own wordsA simple closed curve made up of three or more line segments

CharacteristicsClosed; simple (curve does not intersect itself)2 dimensional; 3 or more line segments

ExamplesSquareTriangleRectangleTrapezoidHexagon

Non-examples

CircleConeRay Pyramid

Polygon

Building Vocabulary: Word Wall Organized collection of words displayed in large letters on wall

Post vocabulary with precise, simple definitions

Effectiveness depends on incorporating word wall into daily instruction

Building Vocabulary: Word Card

Dictionary/Textbook DefinitionA three-sided polygon

Everyday DefinitionThe shape of a yield sign

Diagram, Picture, or Illustration

What the word is NOTCircle, rectangle, square

Adapted from the Michigan Department of Education, Writing Across the Curriculum, 2009

Building Vocabulary: Word Sort

I learned this in elementary

I’ve heard this word

I haven’t learned this yet

SumMultiplicationSubtractionDividendDivisor

EquationExpressionSolutionAlgebra

FunctionLinear equationQuadratic equation

Adapted from the Michigan Department of Education, Writing Across the Curriculum, 2009

Summarizing Mathematics Texts with GIST (Generating Interactions Between Schemata and Text)

Read a portion of text, and write down important ideas and phrases.

Using important ideas, write a summary in 15 words or less.

Continue portion by portion for longer pieces of text.

Write a short summary for the entire selection.

Adapted from the Michigan Department of Education, Writing Across the Curriculum, 2009

Learning Goals Revisited Define mathematical literacy Design instruction for translating across multiple forms of representation in mathematics

Analyze mathematical texts and identify the reading demands students will encounter

Use a variety of teaching tools to support students’ reading

Design instruction for communicating mathematical understanding in writing

Questions?

Thank you!

Julie LearnedJlearned@umich.edu