Ron Eglash
Eglash@rpi.edu
Forthcoming from “Communicator”
(journal of the California Math Council)
Culturally Situated Design Tools in California Classrooms
The field of “ethnomathematics” was introduced by Marcia Ascher and Ubi D’Ambrosio
as a way of discussing the mathematical practices of various cultural groups. Culturally
Situated Design Tools are a suite of web-based interactive applets that allow students and
teachers to explore ethnomathematics through the simulation of cultural artifacts,
including Native American beadwork, African American cornrows, ancient Mayan
temples, Graffiti, and Latin percussion rhythms. The design tools are organized by
cultural group, and their grade levels range from 4 to 12. Each site includes a cultural
background section, a tutorial, the software itself, and a “teaching materials” section of
lesson plans, evaluation instruments, samples of student work, and other support. The
design tools can be integrated into standards-based curricula, including a variety of
specific math topics as well as state and national standards for more general areas such as
technology use and understanding patterns. This paper will discuss some of the work that
has been done specifically with teachers in California, and review the ways these tools
can boost both student interest and performance.
1) Running the design tools on your computer
The design tool homepage is located on the web at http://www.rpi.edu/~eglash/csdt.html.
As long as you are using an Internet Explorer browser, you can use either Macintosh or
Windows. You do not need to download the design tools; they run inside the web
browser. However you may need to update the flash and java “plugins” for the browser.
You can do that by following the links “Click here to get the latest java plugin” and
“Click here to get the latest flash plugin” at the design tool homepage above. Often
schools will place security blocks on their computers that prevent students from
downloading inappropriate software; this will often block access to the plugins as well. If
you can’t get the plugins to download, it probably means you need to get your school’s
technical help to get around this security block.
2) The Virtual Bead Loom: Teaching Cartesian Coordinates though Native
American Design
Eduardo Arismendi-Pardi, Professor of Mathematics at Orange Coast College, has been
instrumental in exposing many California teachers to the design tools. In particular his
student, middle school teacher Adriana Moreno Magallanes, has shown that middle
school students can benefit from use of the Virtual Bead Loom (VBL). Lets take an
overview of the VBL before we examine how Magallanes specifically taught with it.
The opening webpage for the Virtual Bead Loom (VBL) allows users to select from four
categories: Cultural Background, Tutorial, Software, and Teaching Materials. The
cultural background section opens with several examples of four-fold symmetry in Native
American design. Before reading the text, teachers can ask students to look at the designs
and describe them; such discussions offer opportunities to introduce symmetry as a term
and concept. The text describes how four-fold symmetry is a deep design theme in many
Native American cultures, and is evident not only in a wide variety of native arts, but also
indigenous knowledge systems such as base four counting, four-quadrant architecture, the
"four directions" healing practice, etc. A second web page shows how such structures are
analogous to the Cartesian coordinate system. In Navaho sandpainting, for example, we
see the use of vertical and horizontal axes, with paired negative and positive ends.
Finally, the webpage introduces the Native American bead loom as another example in
which we find a Cartesian grid.
The tutorial begins by showing how the VBL simulates a traditional Native American
bead loom, and reviews the various tools. The VBL tools (figure 1) appear to the right of
Figure 1: Virtual Bead Loom
the grid in which designs are created. The simplest tool, for creating or deleting single
beads, is at top. Students enter the X,Y coordinate pairs, select a bead color (or compose
a custom color if none of the standard buttons suits their taste), and press the “point”
button. Other tools include lines, rectangles, triangles and iterative patterns. There are
three categories of goals one can assign to students:
1. Simulation goals: part of the website includes images of traditional beadwork.
Students can select a “goal pattern” using part or all of a traditional beadwork
pattern.
2. Creative innovations: students can design a pattern of their own choosing.
3. Math inquiry: teachers can create design challenges, explore geometric
relationships, visualize algebraic expressions, etc.
Lesson plans and examples using all three categories can be found on the “teaching
materials” part of the website.
Younger students (it has been used successfully with students as low as 3rd grade)
typically begin by working entirely in the +X,+Y quadrant. Those who are just being
introduced to the Cartesian system have the opportunity to discover the concept of
negative coordinates through guided exploration. One of my favorite conversations goes
something like this:
Student: “I want to get beads over here (motions to +X,-Y quadrant).
Instructor: “Try a lower number”
Student: “But I brought Y down to zero and it’s still stuck on this side”
Instructor: “Try a number lower than zero”
Student: “But there isn’t… oh, negative numbers!”
The VBL was created in collaboration with teachers and students at schools serving the
Shoshone-Bannock reservation in Idaho, the Northern Ute reservation in Utah, and the
Onodaga Nation reservation in New York. It has also been used with a wide variety of
other ethnic groups, including white students, with great enthusiasm.
Adriana Magallanes, teaching at a primarily Latino middle school in California, began by
making use of cultural links between Native American heritage and the Latino heritage of
her students. She used the cultural materials on the site to discuss the possibility that
Native Americans had developed an equivalent of the Cartesian coordinate system
previous to European contact. Then she had students take turns at the computers (she
only had 4 computers in the classroom), and had them generate design of their own
choosing. She then assigned them the task of reproducing their computer-generated
designs by hand (figure 2). Finally, she devised an evaluation for their comprehension of
Figure 2: Student work using Virtual Bead Loom
Cartesian graphs, and used the evaluation to compare the performance of two classes, one
that had received instruction using the software, and the other without it. She found
statistically significant higher scores in the class using the software. Again, the “teaching
materials” section of the VBL website includes all of these materials – lesson plan,
evaluation, etc. – for Magallanes as well as for several other teachers. We would
welcome receiving your own teaching materials or samples of student work for posting
on this site.
3) Cornrows Curves, Rhythm Wheels, and Pre-Columbian Pyramids: a brief
overview
Space limitations do not permit detailed discussion of each of the design tools, so this
section will provide a brief overview of the most popular applications.
The Cornrow Curves software (figure 3) allows students to generate simulations of
Figure 3: Cornrow Curves
cornrow hairstyles using transformational geometry. Each braid is represented as multiple
copies of a “Y” shaped plait. In each iteration, the plait is copied, and a transformation is
applied. The series of transformed copies creates the braid. In the above example, we can
see the original plait at top, and each of the copies are successively scaled down, rotated,
and translated (reflection is only applied to whole braids, as in the case where one side of
the head is a mirror image of the other). The website includes an extensive cultural
history of cornrow hairstyles, from their African origins, through the middle passage,
from civil war to civil rights, and finally to the hip-hop era. It also provides a dynamic
tutorial that introduces each mathematical concept used in the simulation, and allows
students to experiment with each concept so that they can develop an intuitive
understanding of its meaning. Finally, the software itself provides students with a library
of different hairstyle images, and a panel of the transformation parameters they
encountered in the tutorial. As in the case of the VBL, students can be assigned the task
of creating a simulation, creating their own designs, or engaging in a math inquiry
exercise. In additional to learning all four geometric transformations, students can also
learn the Cartesian coordinate system (placement of the braids requires use of a Cartesian
grid), iteration, geometric series, and various topics involving the shapes generated
(circles, spirals, etc.).
The Rhythm Wheels software currently offers the use of Latin percussion and hip-hop
beats; plans are underway to offer a version using Native American music as well. The
software allows students to choose the number of beats in a cycle (up to 16), and to fill
each beat with a selection from a library of percussion sounds. They can have up to 3
cycles operating simultaneously, and also have control over the volume of each beat and
the number of times the cycle repeats. Younger students can experiment with
multiplication (what is the total number of beats played if a 3-beat repeats 3 times?), and
ratio (if a 4-beat cycle and an 8-beat cycle are played simultaneously, which one will
finish first?). Older students can use it to discover the concept of Least Common Multiple
(if you have a 3-beat cycle played simultaneously with a 4-beat cycle, how many times
should each repeat so that both stop at the same time?). The cultural background section
of this software currently provides an overview of Latino-Caribbean music; this will be
expanded to other music traditions in the near future.
The Pre-Columbian Pyramids software allows users to create 2 and 3-dimensional
simulations of architectural designs from ancient Mexico. The cultural background
section provides a library of images of pyramids for simulation goals, along with a brief
description of their ancient origins and use. For 2-dimensional simulations (particularly
appropriate for younger users) there is a collection of the relief designs that were etched
on the exterior of some of the structures. These relief designs and the actual pyramids all
have in common an iterative construction method – that is, they can be generated by
specifying an initial base of blocks, and a rule that determines how many blocks are
subtracted or added from each side in each successive layer. Subtracting bricks from all
sides, for example, will create a typical pyramid, while adding bricks can create some of
the over-hanging structures seen in the 2-D reliefs. Again, student activities can include
simulation, creative design, and math inquiry.
4) Conclusion: classrooms at the crossroads of culture and mathematics
Both teachers and students report that there is something special about incorporating
these design tools into their classroom. From our surveys, several features stand out. One
aspect is the contrast to less interactive teaching methods; in particular the creative
aspects of the work. Another is the inclusion of cultural materials. Of particular interest
here is the question of ethnomathematics: do students conclude that these indigenous
societies really had their own forms of math? There is no right answer here – we believe
that it’s thinking about the question that is important for student learning – but we do find
that a majority of students seem convinced that they are seeing math knowledge of some
type, and that it contradicts the stereotypes they had about these cultures as “primitive.”