Culturally Relevant Mathematics Lesson:

Figure 5.13: Culturally Relevant Interdisciplinary Mathematics Lesson Mathematics – Social Studies – Art – Language Arts – Foreign Language

Lesson Plan: Counting with Quipus

Grade Level: 5- 6 7- 8 9 – 10

Topics: Numbers and Operations. Functions and Algebra.

Objectives:

Language Objective

Students experience non-traditional aspects of language and communication of mathematics by observing the symbolic representation (the knots) of number and its place value. Students make conjectures, compare and contrast with traditional numeric systems such as Roman (for example).

Vocabulary:

Number Place Value Digits Function Variable

Ethnomathematics Ethnography Perú Incas Quechua Khipu

Culture and Ideology Objective

Students will connect mathematics with their own cultural and community histories and appreciate the contributions that various cultures and peoples have made to mathematics.

Students will recognize the power of mathematics as an essential analytical tool to understand and potentially change the world.

Mathematics of the Incas The mathematics of the Incas, represent one of the unexplored ancient mathematics. One uncultivated field students will be fascinated to earn about.

Relevance: Together with the Mayans it represents one of the Americas most sophisticated socio-cultural understandings of the new world, a still to be uncovered mathematical knowledge.

1 What do we know about the Incas?

• The Incas lived in present day Peru, southern Ecuador, northern Chile, western Bolivia and western Argentina • First quipus date from • The civilization flourished from 1400-1532 • The empire was conquered by Spain in 1532

2 Architecture

• its insuperable architecture show mysterious characteristics • it is impossible to put a razor blade in between stones in their structures • they did not use mortar

Road System

• mountainous terrain • extensive • primarily for foot traffic because the Inca did not use the wheel

Bridges

Bridge Construction the Incas constructed bridges entire of rope

3 The Khipu

What does Khipu mean?

• the word “Khipu” comes from the Cusco dialect and simply means “knot” • the Spanish word is pronounced the same way but spelled “Quipu” What are Quipus?

• knotted cords used to record data and other information • types of knots indicate value • placement and spacing of knots indicate value • color can identify the meaning of the value

A brief history

• the use of Quipus actually predates the Inca by at least 400 years • they are still used by some people today • they are primarily associated with the Inca because they used them so extensively

Types of knots

• long knot • single knot • figure eight knot

4 Examples of quipus:

• primary cord with secondary cords and a single secondary cord

Why were Quipus used?

• the Incas were highly skilled with rope • the Incas had no written language • quipus were reliable as the records were difficult to lose, alter or destroy

Mathematical Forms of Communication

• Numeric • Arithmetic • Algebraic • Graphic

Numeric and Arithmetic Representation

6 3643 = 3x103 + 6x102 + 4x101 + 3x100

• L = 1000 (long knot) • S = 100 (single knot closest to primary cord) • s = 10 (single knot closer to end of secondary cord) • E = 1 (figure 8 knot)

• 3L + 6S + 4s + 3E

Algebraic Representation

Graphic

4 1000 100 3 10 2 1 1

Exercise: represent the Quipu in three different ways

Numeric and Arithmetic

8 • 1320 = 1x103 + 3x102 + 2x101 + 0x100 • 14 = 1x101 + 4x100 • 27 = 2x101 + 7x100 • 201 = 2x102 + 0x101 + 1x100 • 1562 = 1x103 + 5x102 + 6x101 + 2x100

Algebraic

• Let L = 1000, S = 100, s = 10, E = 1 • L + 3S + 2s • s + 4 • 2s + 7 • 2S + E • L + 5S + 6s + 2E

Graphic

5 1000 4 100 3 10 1 2

Overview: Students create their own Inca quipus and use them for basic mathematical computation.

Objectives:

The students will:

• Design an Inca quipu. • Demonstrate how to use an Inca quipu for mathematical computation.

Related National NCTM Standards:

Mathematics:

• Uses basic and advanced procedures while performing the processes of computation. • Understands that mathematics has been helpful in practical ways for many centuries. • Understands that mathematicians often represent real things using abstract ideas like

9 numbers or lines; they then work with these abstractions to learn about the things they represent.

Materials:

Illustration of a quipu Length of rope Various colored strings, scissors

Procedure:

Comprehensive Input

Students as Inquirers:

Have students help you find a Native Peruvian guest speaker visit your classroom; you may want to choose a family member among your classroom, school, or community. The closer the relation to your students will make a better impact.

Before the visit of a Native Peruvian guest speaker guide students into searching on the web information about the word quipus -Students have an opportunity to research, find and interpret information at their leisure.

- Students feel more comfortable when they are able to connect new knowledge with past learning. - It will be easier for the teacher to scaffold further information.

The teacher may want to recommend that students gather information by exploring “international connections” using web-CT , web chatting, text messaging, you tube, etc configurations. The teacher should guide this connection.

The Teacher’s Lesson:

1) Show the students the illustration of a quipu, pointing out the various colors of string and the knots that indicate numbers in a decimal system. 2) Distribute one length of rope (about 3 feet long) to each student. Knot each end of the rope. Distribute lengths of different colored strings to each student for them to attach to their main quipu rope, all hanging on the same side of the main rope. 3) To represent a number in the ones column, tie knots at the lowest part of the strings, first making a loop of string and passing one end of the string through the loop. Without pulling the string tight, continue holding the loop and repeat passing the end of the string through the loop until you have made a number up to 9. 4) To record in the tens place, make the knots higher up the string. For the hundreds, make the knots even higher up, etc. 5) Ask the students to make their own quipu to record the number of several things Have them keep the numbers and the items they are counting secret from other students. 6) Then ask each student to exchange his or her quipu with a partner and have them try to figure out the numbers (count the knots) and what the items are (the color) that the knots are counting. 7) For more advanced students, ask them to figure out a method of how to use a quipu to record the population of their class or school, the ages of the students and staff members, and what color hair they have. 8) Quipus vs. Paper and Pencil Arithmetic vs. Calculators: Ask students to discuss the advantages and disadvantages of each of these three different ways of counting. Some students may be

10 interested in also researching the abacus and other tools used by different civilizations for mathematical purposes.

Assessment Recommendations:

Before the visit of the Native Peruvian guest speaker guide students into searching on the web information about quipus.

Have students maintain a journal with their interpretation before and after acquiring the information.

Teachers should keep a portfolio of individual student’s progress.

About the concept:

After your definition, explanation and demonstration, ask the students to draw a quipu, design a quipu, and demonstrate the mathematical connection of a quipu.

Ask the students to test each other by exchanging quipus with a partner and ask them to try to figure out the numbers and items being depicted on the quipus.

Extension and/or Adaptation Ideas:

To simplify the quipu for students, the teacher may demonstrate how to make one.

More advanced students may figure out a method of how to use a quipu to show a more complex concept, like population and ages of people in their school, eye color, and who is buying hot lunch in each classroom.

Compare the advantages and disadvantages of the quipu with other counting systems like the abacus.

Bibliography and Possible Sites for Research:

The teacher recommends websites and research information to guide students where to look for information.

Example: • Ascher, M. (1991) Ethnomathematics, Brooks/Cole Publishing Co. • Quilter, J. and Urton G. Eds. (2002) Narrative Threads, University of Texas Press • en.wikpedia.org • http://agutie.homestead.com/files/Quipu_B.htm • www.incasgroup.it • www.crystalinks.com • www.latinamericanstudies.org • www.mamani-inca.com • www.pbs.org

Contributed by Dr. Eliana Rojas, University of Connecticut-Storrs, Neag School of Education, Grant Director, USDOE Title III National Professional Development Program - TEACEP (2002-2007); Principal Investigator, USDOE Title III Grant (2000-2005).