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Understanding Understanding Mathematical Mathematical

Representations in the Representations in the MediaMedia

Mr. EmichMr. Emich

11/6/200811/6/2008

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Understanding Mathematical Representations in the Understanding Mathematical Representations in the MediaMedia Grade: 7Grade: 7 Subject: MathematicsSubject: Mathematics Rationale:Rationale:

Mathematical arguments are often made to support a Mathematical arguments are often made to support a position in the media. It is important as consumers of the position in the media. It is important as consumers of the media that we are able to look critically at the information media that we are able to look critically at the information presented and make informed decisions regarding its presented and make informed decisions regarding its validity. In this project will we delve deeply into a real validity. In this project will we delve deeply into a real mathematical argument made in the media. Using math mathematical argument made in the media. Using math skills we have already developed we will analyze our findings skills we have already developed we will analyze our findings for full understanding. We will look at the bias presented in for full understanding. We will look at the bias presented in the original argument and then use the same data to create the original argument and then use the same data to create a different argument.a different argument.

Trivial Example:Trivial Example: Headline: “Phillies win the World Series in 5 Games”Headline: “Phillies win the World Series in 5 Games” Alternative Headline: “Tampa Bay Rays are the 2Alternative Headline: “Tampa Bay Rays are the 2ndnd best team in best team in

the world”the world”

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Understanding Mathematical Representations in the Understanding Mathematical Representations in the MediaMedia Objectives: Objectives:

Understand that different mathematical Understand that different mathematical representations of the same data and representations of the same data and analysis can present a different message analysis can present a different message to the reader.to the reader.

Understand that mathematical evidence Understand that mathematical evidence needs to be critiqued in terms of its needs to be critiqued in terms of its credibility and usefulness.credibility and usefulness.

Understand the ways in which Understand the ways in which mathematics is used to represent bias in mathematics is used to represent bias in real world contexts.real world contexts.

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Understanding Mathematical Representations in the Understanding Mathematical Representations in the MediaMedia Relevant NYS Learning Standards:Relevant NYS Learning Standards:

Students will organize and consolidate their mathematical Students will organize and consolidate their mathematical thinking through communication.thinking through communication.

Students will communicate their mathematical thinking Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.coherently and clearly to peers, teachers, and others.

Students will analyze and evaluate the mathematical thinking Students will analyze and evaluate the mathematical thinking and strategies of others.and strategies of others.

Students will recognize and use connections among Students will recognize and use connections among mathematical ideas.mathematical ideas.

Students will understand how mathematical ideas interconnect Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.and build on one another to produce a coherent whole.

Students will recognize and apply mathematics in contexts Students will recognize and apply mathematics in contexts outside of mathematics. outside of mathematics.

Students will create and use representations to organize, Students will create and use representations to organize, record, and communicate mathematical ideas.record, and communicate mathematical ideas.

Students will select, apply, and translate among mathematical Students will select, apply, and translate among mathematical representations to solve problems.representations to solve problems.

Students will use representations to model and interpret Students will use representations to model and interpret physical, social, and mathematical phenomena.physical, social, and mathematical phenomena.

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Understanding Mathematical Representations in the Understanding Mathematical Representations in the MediaMedia Materials:Materials:

Poster boardPoster board Color pensColor pens Copies of Word documents:Copies of Word documents:

Ladder of FeedbackLadder of Feedback Final Presentation RubricFinal Presentation Rubric

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Understanding Mathematical Representations in the Understanding Mathematical Representations in the MediaMediaTimingTiming

This project is a part of a portfolio of This project is a part of a portfolio of “Math Outside the Classroom” “Math Outside the Classroom” projects that students will projects that students will work on throughout the year.work on throughout the year.

These projects extend the mathematical skills already These projects extend the mathematical skills already obtained by the students into real world situations.obtained by the students into real world situations.

These projects typically run over the course of a These projects typically run over the course of a week.week.

Students will be given the task of looking for Students will be given the task of looking for mathematics in the media on a Monday. By the mathematics in the media on a Monday. By the following Monday they need to have their media following Monday they need to have their media selection ready.selection ready.

Groups will be formed on Monday. On Wednesday Groups will be formed on Monday. On Wednesday group peer review will occur.group peer review will occur.

Class Presentations will take place on Friday.Class Presentations will take place on Friday.

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Understanding Mathematical Representations in the Understanding Mathematical Representations in the MediaMedia Overall Project Procedures:Overall Project Procedures:

Individually students will search the media Individually students will search the media for a mathematical argument, that they for a mathematical argument, that they feel they can re-represent to support a feel they can re-represent to support a different position. In class, students will different position. In class, students will team into groups of 3 and discuss the team into groups of 3 and discuss the media items they discovered. As a group media items they discovered. As a group they will choose one item and develop a they will choose one item and develop a draft of a poster that shows the original draft of a poster that shows the original argument and the new argument, using the argument and the new argument, using the same data. Pairs of groups will explain same data. Pairs of groups will explain their posters to each other. The pairs of their posters to each other. The pairs of groups will give feedback to each other. groups will give feedback to each other. Groups can incorporate feedback into their Groups can incorporate feedback into their final poster. Finally groups will present final poster. Finally groups will present their posters to the class. their posters to the class.

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Lesson ProceduresLesson Procedures Day 1Day 1

Introduction to ProjectIntroduction to Project Work through example from: Work through example from:

http://www.nytimes.com/2008/11/06/business/06pay.htmlhttp://www.nytimes.com/2008/11/06/business/06pay.html?_r=1&scp=1&sq=wall%20street%20bonus&st=cse&oref?_r=1&scp=1&sq=wall%20street%20bonus&st=cse&oref=slogin=slogin

Headline “Wall Street Bonuses Plummet 70%”Headline “Wall Street Bonuses Plummet 70%” Work through details if bonus was $68.5 million, as Work through details if bonus was $68.5 million, as

reported in article (bonus is still over $20 million)reported in article (bonus is still over $20 million) Introduce and review assessment rubricIntroduce and review assessment rubric Questions from classQuestions from class Form class into groups of threeForm class into groups of three Groups meet to discuss their media findingsGroups meet to discuss their media findings

Groups need to select which media finding they will Groups need to select which media finding they will chose for their project by the end of the periodchose for their project by the end of the period

Exit slip: Write a brief description the strengths and Exit slip: Write a brief description the strengths and weakness of your media selection.weakness of your media selection.

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Lesson ProceduresLesson Procedures Day 2Day 2

Questions from classQuestions from class Remind students that they will meet in pairs Remind students that they will meet in pairs

of groups tomorrow to peer assess posters of groups tomorrow to peer assess posters Introduce and review Ladder of Feedback Introduce and review Ladder of Feedback Groups meet to work on developing their Groups meet to work on developing their

projectproject Groups need to work out their mathematical Groups need to work out their mathematical

argument by the end of the periodargument by the end of the period Groups meet to work on developing a draft of their Groups meet to work on developing a draft of their

poster, keeping in mind the assessment rubricposter, keeping in mind the assessment rubric Draft has to be ready for review tomorrow by the Draft has to be ready for review tomorrow by the

end of classend of class

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Lesson ProceduresLesson Procedures Day 3Day 3

Questions from classQuestions from class Pair up groups for peer assessmentPair up groups for peer assessment Remind students to use the Ladder of Remind students to use the Ladder of

FeedbackFeedback Pairs of groups meet to perform peer Pairs of groups meet to perform peer

assessment, using the Ladder of assessment, using the Ladder of FeedbackFeedback

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Lesson ProceduresLesson Procedures Day 4Day 4

Questions from classQuestions from class Remind students tomorrow is Remind students tomorrow is

presentation daypresentation day Groups convene to produce final posterGroups convene to produce final poster Final poster must be completed by the Final poster must be completed by the

end of the periodend of the period

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Lesson ProceduresLesson Procedures Day 5Day 5

Questions from classQuestions from class Groups present final posters to classGroups present final posters to class Students choose one project that is Students choose one project that is

presented to assess using the rubric. presented to assess using the rubric. Write up of assessment to be handed Write up of assessment to be handed in at next class.in at next class.

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Understanding Mathematical Representations in the Understanding Mathematical Representations in the MediaMedia Assessments:Assessments:

Individually students will write a one to Individually students will write a one to two page paper describing their media two page paper describing their media selection and their thoughts on the selection and their thoughts on the strengths and weaknesses of it for this strengths and weaknesses of it for this assignment.assignment.

After the groups make their math in the After the groups make their math in the media selection each student will write a media selection each student will write a one to two page paper covering:one to two page paper covering:

What are the strengths of the argument you What are the strengths of the argument you have to present? have to present?

What are the concerns you have about the What are the concerns you have about the argument you have to present?argument you have to present?

These concerns must be discussed during the group These concerns must be discussed during the group work on the project.work on the project.

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Understanding Mathematical Representations in the Understanding Mathematical Representations in the MediaMedia Assessments:Assessments:

Students will be provided with “Ladder of Students will be provided with “Ladder of Feedback” forms to be used during peer Feedback” forms to be used during peer review. These forms are to be filled out review. These forms are to be filled out and handed in.and handed in.

Before final presentations each student Before final presentations each student will be given copies of the rubric. will be given copies of the rubric. Students will assess presentations and Students will assess presentations and write up their assessment on one project write up their assessment on one project to be handed.to be handed.

Final presentations will be evaluated by Final presentations will be evaluated by the teacher according to the rubricthe teacher according to the rubric..

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Understanding Mathematical Representations in the Understanding Mathematical Representations in the MediaMedia Ladder of Feedback Form:Ladder of Feedback Form:

Feedback For:

1. Clarify: Are there aspects that 1. Clarify: Are there aspects that you don’t believe you have you don’t believe you have understoodunderstood

2. Value: What do you see that you 2. Value: What do you see that you find to be particularly impressive, find to be particularly impressive, innovative, strong?innovative, strong?

3. Offer Concerns: Do you detect 3. Offer Concerns: Do you detect some potential problems or some potential problems or challenges? Do you disagree with challenges? Do you disagree with something?something?

4. Suggest: Do you have 4. Suggest: Do you have suggestions on how to address the suggestions on how to address the concerns you identified during the concerns you identified during the last step?last step?

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Understanding Mathematical Representations in the Understanding Mathematical Representations in the MediaMediaLadder of Feedback Form Explanation:Ladder of Feedback Form Explanation: 1. Clarify1. Clarify

Are there aspects that you don’t believe you have understood?Are there aspects that you don’t believe you have understood? Ensure that you're clear about what your colleague hopes to achieve by asking some Ensure that you're clear about what your colleague hopes to achieve by asking some

questions or stating any assumptions you've made (i.e., “I wasn’t sure if you meant that questions or stating any assumptions you've made (i.e., “I wasn’t sure if you meant that …, but that’s what I assumed, so now you can understand where my feedback is coming …, but that’s what I assumed, so now you can understand where my feedback is coming from.”) from.”)

2. Value2. Value What do you see that you find to be particularly impressive, innovative, What do you see that you find to be particularly impressive, innovative,

strong?strong? Valuing builds a supportive culture of understanding and will help your colleague to Valuing builds a supportive culture of understanding and will help your colleague to

identify strengths in their work that they might not have recognized otherwise identify strengths in their work that they might not have recognized otherwise Valuing reminds your colleague of the parts of his/her work that should be preserved Valuing reminds your colleague of the parts of his/her work that should be preserved

when making improvementswhen making improvements Expressing your appreciation for others and their ideas is fundamental to the process of Expressing your appreciation for others and their ideas is fundamental to the process of

constructive feedbackconstructive feedback Stressing the positive points of the work and offering honest compliments sets a Stressing the positive points of the work and offering honest compliments sets a

supportive tonesupportive tone 3. Offer Concerns3. Offer Concerns

Do you detect some potential problems or challenges? Do you disagree Do you detect some potential problems or challenges? Do you disagree with something? with something?

Share your concerns, not as derisive accusations or abrasive criticisms, but as honest Share your concerns, not as derisive accusations or abrasive criticisms, but as honest thoughts (i.e., "Have you considered . . ."; "What I wonder about is. . ."; "Perhaps you thoughts (i.e., "Have you considered . . ."; "What I wonder about is. . ."; "Perhaps you have thought about this, but . . .".)have thought about this, but . . .".)

4. Suggest4. Suggest Do you have suggestions on how to address the concerns you identified Do you have suggestions on how to address the concerns you identified

during the last step? during the last step? Help your colleague make improvements by sharing your ideas on how he/she might Help your colleague make improvements by sharing your ideas on how he/she might

revise their work (of course, there is no guarantee that your colleague will use the revise their work (of course, there is no guarantee that your colleague will use the suggestions -- suggestions are not mandates, after all!)suggestions -- suggestions are not mandates, after all!)

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Understanding Mathematical Representations in the Understanding Mathematical Representations in the MediaMedia Final Presentation Rubric:Final Presentation Rubric:

CATEGORY 4 3 2 1

Picked appropriate mathematical argument from media

Picked an article which made a mathematically based argument. Can fully explain what position is being conveyed.

Picked an appropriate article, but lacked some clarity about what position is being conveyed.

Picked an appropriate article, but is unclear about what position is being conveyed.

Did not pick an article with a mathematical argument.

Understands mathematical concepts of original argument

Able to articulate mathematical concepts used in argument Able to articulate data source of original argument

Able to articulate mathematical concepts but is not totally clear on data source or vice versa.

Partially able to articulate mathematical concepts used in argument Partially able to articulate data source of original argument

Not able to articulate mathematical concepts used in argument Not able to articulate data source of original argument

Made mathematical connections to reposition data into a different position.

Able to create a convincing mathematical argument based on the same data as the original argument.

Able to create a convincing mathematical argument but had to extend data.

Used the same data, but mathematical argument while correct, did not support a different position.

Not able to create a convincing mathematical argument based on the same data as the original argument.

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Understanding Mathematical Representations in the Understanding Mathematical Representations in the MediaMedia Final Presentation Rubric (continued):Final Presentation Rubric (continued):

CATEGORY 4 3 2 1

Group creates a poster that is accurate, clear and presents a convincing conclusion

Explanation is detailed and clear. Grammar and spelling correct. Diagrams and/or sketches are clear and greatly add to the understanding.

Explanation is clear. Minimal spelling and grammatical errors. Diagrams and/or sketches are clear and easy to understand.

Explanation is a little difficult to understand, but includes critical components. Several spelling and grammatical errors. Diagrams and/or sketches are somewhat difficult to understand.

Explanation is difficult to understand and is missing several components OR was not included. Many spelling and grammatical errors. Diagrams and/or sketches are difficult to understand or are not used.

Mathematical Concepts

Explanation shows complete understanding of the mathematical concepts used.

Explanation shows substantial understanding of the mathematical concepts used.

Explanation shows some understanding of the mathematical concepts.

Explanation shows very limited understanding of the underlying concepts.

Mathematical Reasoning

Uses complex and refined mathematical reasoning.

Uses effective mathematical reasoning

Some evidence of mathematical reasoning.

Little evidence of mathematical reasoning.

Mathematical Errors

90-100% of the steps have no mathematical errors.

Almost all (85-89%) of the steps have no mathematical errors.

Most (75-84%) of the steps have no mathematical errors.

More than 75% of the steps have mathematical errors.

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Student Portfolio – Math Outside of the Student Portfolio – Math Outside of the ClassroomClassroom

Statement of purpose:Statement of purpose: You will be developing a showcase You will be developing a showcase portfolio over the course of this year. This portfolio will portfolio over the course of this year. This portfolio will support your learning and demonstrate growth in the area of support your learning and demonstrate growth in the area of “Math Outside the Classroom.” The contents of this working “Math Outside the Classroom.” The contents of this working portfolio will help you and your classmates answer the portfolio will help you and your classmates answer the question: “When am I ever going to use this stuff?” There question: “When am I ever going to use this stuff?” There will be plenty of opportunity for you and your classmates to will be plenty of opportunity for you and your classmates to assess your portfolio work as we progress throughout the assess your portfolio work as we progress throughout the year. There will always be opportunity for you to modify and year. There will always be opportunity for you to modify and enhance your work, based on your feedback and feedback enhance your work, based on your feedback and feedback from your classmates. The contents of this portfolio will be from your classmates. The contents of this portfolio will be used to develop a project at the end of the year that used to develop a project at the end of the year that demonstrates what you learned and how your thinking demonstrates what you learned and how your thinking changed regarding “Math Outside the Classroom”, changed regarding “Math Outside the Classroom”, throughout the year. The audience for your portfolio will be throughout the year. The audience for your portfolio will be yourself, classmates, your teachers and your parents. yourself, classmates, your teachers and your parents. Go back to: Go back to: Understanding Mathematical Representations in the Media