The Shapes of Our WorldExperimenting With the Language of GeometryAuthor(s)Jennifer   Rittner, The New York Times Learning Network Tanya   Yasmin Chin, The Bank Street College of Education in New York City Grades: 6-8, 9-12Subjects: Mathematics, Science, Social StudiesInterdisciplinary Connections

Overview of Lesson Plan:In this lesson, students play a game of charades as an experiment in non-verbal communication. They then create maps with directions that demonstrate their ability to utilize shapes and spatial relationships in a practical context. Their learning is culminated in a written critical essay about the universality of human understanding.Review the Academic Content Standards related to this lesson.Suggested Time Allowance:1 hourObjectives:Students will:1. Play charades in order to discover if they use any common non-verbal signs or symbols.2. Learn about an experiment in cognition by reading and discussing the article "Mastering the Geometry of the Jungle."3. Create maps and directions that use geometric concepts.4. Write a critical reflection paper that explores the idea of the universal concepts in human thinking.Resources / Materials:-pens/pencils-several small sheets of paper as outlined in the Warm-Up-container for holding the papers (such as a box or hat)-classroom board-colored markers (enough for each pair)-transparent or masking tape (enough for each flag)-colored paper for making squares (each student will need two squares of matching color)-copies of the article "Mastering the Geometry of the Jungle," found online at http://www.nytimes.com/learning/teachers/featured_articles/20060124tuesday.html (one per student)Activities / Procedures:1. WARM-UP/DO-NOW: Before class, write the titles of books, movies or songs, down on several small pieces of paper, noting one title per slip. Include at least one of the following concepts in your titles: time, a number, a shape or a direction in space. Place the papers into a container. At the beginning of class, explain to students that they will play a game of charades. Invite volunteers, one at a time, to choose from the container of titles and perform the game. Complete at least two rounds. At the successful completion of each round, have the student write the title on the board for later reference. After the last round, ask students the following, "Did you understand the symbols presented by each of the performers? What instructions were the hardest to understand? Which were the easiest?" Prompt them, if necessary, to consider how time, space and numbers were presented. Ask students if they think there is a common way in which people understand these concepts.

2. As a class, read and discuss the article "Mastering the Geometry of the Jungle" (http://www.nytimes.com/learning/teachers/featured_articles/20060124tuesday.html), focusing on the following questions:a. Where do the Munduruku live?b. For what geometric shape does their language have a word?c. How do they identify directions?d. What have researchers discovered in their studies of the Munduruku?e. What specific "abilities that are the center of human thinking" were the researchers studying?f. How did they test their theory and who, other than the Munduruku did they use as subjects in their experiment?g. How many test groups participated in the test and how did those groups compare in completing the geometry test?h. What was the map test designed to demonstrate, and what was the outcome?i. What earlier study was done to suggest that geometry may be a universally-understood concept?j. Who conducted that earlier study; who was the subject, and why might he have been chosen?k. What subtle point does Stanislas Dehaene make about interpreting the data they have collected?l. How and why does he distinguish innate ability and innate knowledge?m. What does Dr. Dehaene believe the data suggests in terms of human thinking across cultures?

3. Explain to the class that they will create maps and directions for traveling from point "A" to point "B." This exercise may be conducted in the classroom (with the desks cleared away), in an auditorium, outdoors, or in any other space that can accommodate your students. To begin, arrange class into pairs and give each pair two sheets of white paper, tape, markers, and two squares of matching color that they will label "Point A" and "Point B." Their goal is to create a non-linear path from Point A to Point B that utilizes at least four geometric principles (and therefore at least four steps), such as a line, arc, right angle, acute angle, square, etc. You should instruct them that Point A and Point B may be as far or as close as they choose as long as the directions adhere to the parameters set above. Write the following directions on the board:

Step 1: Identify your start (Point A) and finish (Point B) points and place the appropriate flag at each location. (Instruct them that they may use tape to secure it to the area.)Step 2: Decide how you will get from Point A to Point B by walking a path. Make sure you include at least four geometric concepts. Be creative!Step 3: Draw your directions on one sheet of paper. (Instruct them that on this map, they may only use words that suggest the number of steps, but no other indicators of directionality, such as degrees of a turn.Step 4: On the second sheet of paper, write out step by step instructions for traveling from Point A to Point B following the path you created in your drawing. (Instruct them that they should use creative descriptions to identify directionality, window-ward and hall-ward, as opposed to east or west.)

Once the maps are complete, have pairs exchange their maps. Then have the pairs follow the instructions, first from the drawing, then from the written directions. Did students make it

successfully from Point A to Point B both times? Have them give a brief report of their experience using each set of directions.

If time permits, have pairs identify each of the geometric concepts on the drawn map they were given. Are they able to identify all of the concpts? To what degree, if at all, did knowing or not knowing the concept affect their ability to follow the drawn directions? How about the written directions? To what degree did understanding the creative descriptions affect their ability to follow the written directions?

4. WRAP-UP/HOMEWORK: The article quotes Elizabeth Spelke as saying, "Across cultures that live extremely different lives, we see common foundational sets of abilities." Write a critical reflection (2-3 pages) that explores this idea, considering the following questions (prepared as a handout for students):-How might you communicate concepts of space (giving directions) or time to someone who speaks a different language?-How do people use common symbols (such as shapes and numbers) to supplement language in communicating across cultures?-Why might it be important or interesting to believe that all human beings share a common ability to understand the world that goes beyond language?

Further Questions for Discussion:-To what degree, if at all, do you believe that all human beings share a common ability to communicate without language?-To what degree, if at all, do you recognize or consider geometric concepts outside of the math classroom? Do you participate in any activities that use or require an understanding of geometry (such as sports or other games, sewing, painting, etc.)?-To what degree, if at all, do you think you need to know the mathematical or scientific foundations of something in order to be able to use it in everyday life?Evaluation / Assessment:Students will be evaluated based on participation in the opening exercise, thorough preparation of their maps and participation in the follow-up discussion, and thoughtful completion of their essays.Vocabulary:indigenous, geometric, consistently, foundational, illustrating, linguist, isosceles, data, orient, mentally, symbols, extract, universal, elicited, contention, innate, extent, interact, navigate, spatial, predisposed, internalize, representations, constructions, qualitativelyExtension Activities:1. Identify the geometric concepts for the objects you use from day to day, such as, chairs, tables, pencils, and mugs. Why are these objects shaped and angled the way they are? What are the structural purposes for the shapes of things? Create a model of the object (use toothpicks, tongue depressors, paper, cotton, glue and whatever other materials are available to you) and prepare directions that identify the specific geometric characteristics that compose the object. For additional consideration, conduct an experiment in which you alter the shapes and angles a little at a time until the structure becomes unsound. Record the details of your experiment and write a report that addresses the geometric principles at play.

2. Create a map of your route to and from school that could be used by someone who does not speak your language (or cannot read). To begin, identify your exact start and end points. What geometrical concepts will you utilize to instruct someone on the correct route? What symbols can you use to point someone in the right direction? In addition to the physical geometry of the route, what other identifiers will you use to aid travelers (such as showing landmarks)? Make the maps specific but be creative. Submit final work in a later class and consolidate them into an atlas called "Our Routes."

3. Read about a theorist in cognition studies such as Carl Jung, John Berger, Lev Vygotsky, Jean Piaget or Roland Barthes. What theories about cognition and human intelligence did he consider in his research? How and with what subjects did he test his theories? What were the outcomes of the research? How do you feel about the theory? Write a report that responds to these questions.

4. How do people represent universal concepts, such as time, without words? Identify and collect examples of the various ways people have kept track of time in the short and long term (second, hours, days, weeks, etc.). How did each of those methods address the needs of that particular culture or community? Are those methods understandable today? Try to use that method of time-keeping for an identified period of time (a class period, a day, a week) and report back to the class on your experience.

5. How do scientists identify subjects for research into human behavior? Identify a research study that used human subjects. What was the study? Who were the subjects? How were they used? What were the outcomes? Was there any controversy concerning the use of these subjects? Write a detailed report of your findings and include an analysis that addresses your feelings about the use of human subjects in research.

Interdisciplinary Connections:Fine Arts - Review the work of artists such as Mondrian, Miro, Rothko, Calder, Chagall, DaVinci, etc. Identify the various shapes and angles they use in their work. Is the work representational or abstract? How does that factor into the way geometry is used in the work? How do these artists compare in terms of the way they use geometry and the prominence of strict geometrical forms in their work? Based on what you saw, create a painting or drawing that uses geometric concepts. Present final art works in a class exhibition.

Global Studies - Identify the symbols used in different parts of the world, such as those to identify directions (up, down, east, west), locations or services (hospital, ladies room, pharmacy, police) or other types of critical information (such as danger, incline, merge). Prepare a glossary of the country's symbols and meanings. Are any of the symbols recognizable in your own country or community? Are there any that might not be recognized in your country or that might be used to communicate different information? If so, identify those alternative meanings in your glossary. Compile glossaries into a larger book to submit to the library as a reference.

Teaching with The Times - Read the Science section of The Times every week for one month and clip articles related to cognition theories and research. At the end of the month, write a reflection that summarizes the main issues that arise. What is the theory? Who are the researchers? Where is the research taking place and with what subjects? What do researchers

hope to discover from their work? How might that affect your life and the lives of your friends and family?

Other Information on the WebAmerican Association for the Advancement of Science (http://www.aaas.org/) offers some information about the Munduruku experiment, including visual examples of the tests conducted.

Science Magazine (http://www.sciencemag.org/) publishes a report on the Munduruku study in the January 20, 2006 issue (full text available with subscription).

Science for Kids (http/://www.eurekalert.org/scienceforkids/) also offers information and images related to the research with the Munduruku.

Stanislas Dehaene ( http://www.unicog.org/main/pages.php?page=Stanislas_Dehaene) offers some links to his research.

Academic Content Standards:This lesson plan may be used to address the academic standards listed below. These standards are

drawn from Content Knowledge: A Compendium of Standards and Benchmarks for K-12 Education: 3rd and 4th Editions and have been provided courtesy of the Mid-continent Research for Education and Learning in Aurora, Colorado.Grades 6-8 Mathematics Standard 1- Uses a variety of strategies in the problem-solving process. Benchmarks: Understands how to break a complex problem into simpler parts or use a similar problem type to solve a problem; Uses a variety of strategies to understand problem-solving situations and processes; Understands that there is no one right way to solve mathematical problems but that there are different methods; Formulates a problem, determines information required to solve the problem, chooses methods for obtaining this information, and sets limits for acceptable solutions; Represents problem situations in and translates among oral, written, concrete, pictorial, and graphical forms; Constructs informal logical arguments to justify reasoning processes and methods of solutions to problems; Uses a variety of reasoning processes Mathematics Standard 9- Understands the general nature and uses of mathematics. Benchmarks: Understands that mathematics has been helpful in practical ways for many centuries; Understands that mathematicians often represent real things using abstract ideas like numbers or lines--they then work with these abstractions to learn about the things they represent Science Standard 14- Understands the nature of scientific knowledge. Benchmarks: Knows that all scientific ideas are tentative and subject to change and improvement in principle, but for most core ideas in science, there is much experimental and observational confirmation; Understands that questioning, response to criticism, and open communication are integral to the process of science Geography Standard 4- Understands the physical and human characteristics of place. Benchmarks: Knows the human characteristics of places; Knows the physical characteristics of places; Knows the causes and effects of changes in a place over time

Grades 9-12 Mathematics Standard 1- Uses a variety of strategies in the problem-solving process.

Benchmarks: Uses a variety of strategies to understand new mathematical content and to develop more efficient solution methods or problem extensions; Constructs logical verifications or counter examples to test conjectures and to justify algorithms and solutions to problems; Understands connections between equivalent representations and corresponding procedures of the same problem situation or mathematical concept; Understands the components of mathematical modelingMathematics Standard 9- Understands the general nature and uses of mathematics. Benchmarks: Understands that mathematics is the study of any pattern or relationship, but natural science is the study of those patterns that are relevant to the observable world; Understands that mathematics began long ago to help solve practical problems; however, it soon focused on abstractions drawn from the world and then on abstract relationships among those abstractions; Understands that theories in mathematics are greatly influenced by practical issues--real-world problems sometimes result in new mathematical theories and pure mathematical theories sometimes have highly practical applications; Understands that mathematics provides a precise system to describe objects, events, and relationships and to construct logical arguments Science Standard 14- Understands the nature of scientific knowledge. Benchmarks: Knows ways in which science distinguishes itself from other ways of knowing and from other bodies of knowledge; Knows that scientific explanations must meet certain criteria to be considered valid; Understands how scientific knowledge changes and accumulates over time; Knows that from time to time, major shifts occur in the scientific view of how the world works, but usually the changes that take place in the body of scientific knowledge are small modifications of prior knowledgeGeography Standard 4- Understands the physical and human characteristics of place. Benchmarks: Knows how social, cultural, and economic processes shape the features of places; Understands why places have specific physical and human characteristics in different parts of the world

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