Putting A Lid On It – One Example Of Ethnomathematics Becoming Pedagogical
Action
Introduction
I have long been interested in the power of “objects to think with”, be they
high-tech logo turtles (Orey, 1984) or easy to find but equally powerful
hands-on manipulatives (Fuys & Tischler, 1979; Van de Walle, 1997). As
well, I have been keenly interested in developing ways that schools without
a great deal of financial resources can adapt methods that allow all children
to be successful. This is especially important in Latin American public schools.
Equally exciting for me is to see interest in children and adults in something
as common, mundane and simple as found in our daily environment such as a
jar lid. To be witness to groups of learners engaged in the development of
their own ideas, problems and explorations with simple tools is a privilege.
A number of years ago I came across an activity that used lids to teach numerous
mathematical concepts (Langbort, 1988). This discussion owes much to the
initial ideas that were presented in that article. I have since then taken
a few of Prof. Langbort’s ideas and adapted them to my realities with pre-service
teacher education and as an in-service tool for teachers in California, Texas,
New Mexico and in Brazil (São Paulo, Paraná and Rio Grande
do Sul), and with numerous children in the United States and Brazil.
The importance of students and teachers becoming involved in rich explorations
using a variety of computational and estimating techniques has been recognized
as an important aspect to mathematical reform (Parker, 1993). It is my opinion
that it is an important aspect of ethnomathematics from the view of practical
concrete pedagogical action. Number sense can be both developed and enhanced
by assisting learners in making connections to real-world contexts. Teachers
who provide opportunities that encourage their students to actively engage
and interact mathematically using more than just the memorization of simple
arithmetic and traditional algorithms are empowering their students to become
critical thinkers and problem solvers. This is a basic and fundamental point
of “ethnomathematics as pedagogical action”. Students as well need opportunities
to practice problem solving by actively developing and solving their own
mathematical problems. They will do this even if they are not given this
experience in school – witness the engagement of children the world over
with soccer and commuter games.
I believe that one effective way to do this is by using informal concrete
examples or materials. Given the current emphasis on “back to basics”, “new
and rigorous standards”, and “assessment and evaluation”, investigations
using mathematical modeling are even rarer for children of elementary school
age to engage in at the moment. An ethnomathematical perspective offers support
for activities that allow all students to explore multidimensional contexts
from a mathematics viewpoint. As well, it allows for open-ended problem solving
which often involves more than just one 50-minute period for research and
exploration. Though often centered on one theme or event, an ethnomathematically-based
activity is by its very nature cross-disciplinary.
The intellectual activity of the child must be allowed to come as close as
it can to that done by people using mathematics in the real world (D’Ambrosio,
1996; 1998). Like all empowered users / learners of mathematics, learners
must be given experiences to enable them to model and use mathematics – this
is the essence of constructivism, as well.
An ethnomathematical perspective is more than just one activity immersed
in area of memorization and drill. The premise of this discussion is to demonstrate
one activity that demonstrates a practical application that shows mathematics
as a creative and spontaneous discipline. What it hopes to disclose is the
very essence of pedagogical work for both the teacher and learner, where
the real-world reality of the learner can be taken into account by the teacher,
curriculum and school. Asking children to observe, to truly pay attention,
to make use of real-life based experiences as they learn to observe and interact
in the world around them is a very important opportunity to make connections
for children to mathematics. It also forms the fundamental basis to the scientific
method of inquiry. If children are asked to resolve problems related to real-life
phenomena, we can show them how we can use data they gather from their own
reality. Students can then learn how to practice and learn algorithms from
with-in this context, a context they themselves helped create. They can easily
learn to make sense of statistics from the data they themselves create from
their own interaction with phenomena. Children must be given opportunities
to actively participate in constructing their own reality. To waste this
opportunity – this seemingly innate energy and enthusiasm does not make sense,
nor seem wise to me. We must use this natural desire to assist learners to
elaborate mental structures that allow them to construct and master mathematical
knowledge on their own terms, not just ours, or that of an imposed curriculum
standard.
Objectives
The student will use a “sort and categorize” strategy with a large number
of lids to practice basic addition and subtraction algorithms. They will
develop and solve questions arising from the data they collect after sorting
and categorizing the lid collection.
Materials
- A file box of household lids (100+ is best) of various
sizes, shapes, and colors
- Butcher paper or chalkboard
- Markers / chalk
- Unifix cubes
- Brown lunch bags
Preparing the Investigation
A branch diagram is traced on the chalkboard (use the overhead projector
to trace accompanying diagram). The lids are then spilled on the classroom
floor in a large open area in front of the blackboard.
Structuring the Investigation
The lids are to be sorted according to a variety of attributes (Langbort,
1988, p.22). Which include but are not limited to:
- Size (big and not big, small and not small);
- Shape (round and not round, snap on and not snap on);
- Color (red and not red);
- Information on it relating to its use (peanut butter
and not peanut butter lids)
- Language (Portuguese and not Portuguese)
- Numbers (has numbers on it or does not have numbers on
it)
Children write the categories on the branches of the tree diagram. The teacher
models it by beginning with on attribute: plastic and not plastic and making
the kids sort the entire pile into these two attributes. Students are then
asked to continue sorting into categories of their own choice, making sure
each branch is always an and/or or if/not statement, and labeled clearly
on the board. Depending on the age of the kids and the number of lids in
the box, we go until most of the branches are filled or until a few categories
have a very small number of lids in each. If students need practice with
large numbers, the sorting can stop when groups of lids are in their teens
(i.e. 18 lids end up in a category like: plastic, no number, red, peanut
butter lids).
After the lids are sorted, and before labeling the bags, kids are asked and
given time to ponder “which has more plastic or not plastic?” Then all the
lids are sorted again; this time into the lunch bags that are taped to the
chalk tray underneath the tree diagram aligned with heir particular attributes.
This portion can be used as a learning / activity station for younger children.
When the lids are sorted into each bag, children work in pairs to count the
lids, label the bag and make a Unifix tower to represent the number of lids
found in each bag. Graphs are then constructed on the chalkboard tray using
the Unifix towers representing the number of lids in each category. Children
are then asked again to explain (and then write about) “which has more” plastic
or not plastic? Three good questions that might be used here are:
How do you know you were correct?
How did you go about solving the problem? And
What is the answer?
Other related guided questions from Langbort (1988) include:
Which category has the most lids?
How many more of __ do we have than __?
How many __and__ do we have in all?
How many more __ lids do we need so that we will have just as many as __?
Once kids know how many lids are in the collection further questions can
be posed as generated by the students themselves. The questions posed focus
on how many lids might:
- Fill the classroom? The cafeteria? The gymnasium?
- Cover the floor? The gymnasium? The basketball court?
The soccer field?
- If every household in our school threw away “x” lids
every week, how many are thrown away in a month? A year? How much space does
something so small take up in a landfill?
- If we lined our lids up in a straight line, how many
does it take to cross the playground? How many would it take to go to São
Paulo (posed in Campinas).
Kids form groups around the questions they generate, and are encouraged to
follow this sequence in mathematically modeling the answer to their question:
- Break a problem situation into manageable parts;
- Create a hypothesis;
- Test the hypothesis;
- Correct the hypothesis;
- Make transference and generalizations to their own reality
(Orey & Rosa, in press).
Extending the Investigation
I have had the opportunity to use these activities in numerous locations
in the United States and Brazil. When I included a collection of lids that
I “imported” from California, one obvious category immerged: English and
non-English lids. As well, some very heated discussions arose within the
teachers related to what categories the lids need to be sorted into. Overall,
in both countries children are far more flexible that the adults at this
task. Something that time and time again makes me smile. For example, there
are lids that can easily be categorized for sorting into “dirty and not dirty”
attributes. Adults make many comments, often wishing to throw these lids
out, whereas universally children quite easily make new categories.
Where is the mathematics inherent in this work? Categorization, averaging,
graphing, counting and reflecting on relationships between quantities are
basic to most state and national standards. Students learn to categorize
using a binary system of categorization, something inherent in sophisticated
computer systems. Students also do something that is very important for an
ethnomathematical perspective. They generate their own numbers form within
their own context. The numbers generate themselves by the sort, and are constructed
by the random use of something very mundane – lids. The numbers and problems
that immerge from this exploration are those that are used to rehearse traditional
algorithms, and to develop, write and solve problems that teachers and student
wrote themselves. This is something divergent from the growing movement to
proscribe and script curriculum in mathematics classroom presenting the type
of learning experience for children that I feel is an anathema to constructivism
and empowering mathematics programs.
When this manipulative is extended to explore π by taking a random selection
of lids from the class collection and performing measurements, they further
are able to see the mathematics in their own unique and real world context.
The hope being, and certainly the teacher is encouraged to assist them in
remembering what π is whenever round objects are seen, This is so much easier
for both the teacher and student when using something that they can really
see on a daily basis. It is the essence of empowerment, and empowerment is
the true essence of ethnomathematics.
<I want to write more here but I am out of space for the outline – I will
elaborate if given permission – the gist being kids explore circles using
sting, and measuring diameter and circumference, I have done this with very
young kids>
Concluding Comments
Objects to think with are inherent to the development of pedagogically sound
ethnomathematical curriculum experiences. After all, much of the ethnomathematical
research being done today focuses on the use and construction of concrete
objects unique to a given culture. It makes sense to begin looking at applications
of materials that are found in our day to day life as well. There is room
for work that asks how we can achieve the aims and ideas as outlined by “D’Ambrosio
and the countless researchers (Powell & Frankenstein, 1997; Zaslavsky,
1996) of mathematical experiences for learners using ethnomathematics. In
an increasingly globalized world, where mass marketing and information overload
creates the seeming “necessity” for “teacher proof” materials and scripted
curriculum, it is incumbent upon ethnomathematicans to be at he forefront
of the fight that encourages teachers to retain control of their own knowledge.
A movement that seeks to empower students needs to remind us that this empowerment
brings knowledgeable, empowering, trusting, and supportive teachers.
References
Fuys, D. & Tischler, R. (1979). Teaching Mathematics in the Elementary
School (2nd Edition). Boston: Little, Brown & Co.
Langbort, C. (1988, November). “Jar Lids - An unusual math manipulative”.
The Arithmetic Teacher. (Vol36) no 3 pp.22-25.
Orey, D. (2000). Chapter. " Geometry of the Tipi and Cone: Using Mathematical
Modeling as Applied Ethnomathematics" in Mathematics Across Culture: the
History of Non-Western Mathematics. (Selin, H. Ed.). Dordrecht, Netherlands:
Kulwer Academic Publishers.
Orey, D. (1998, January). "Mathematics for the 21st Century." In: Teaching
Children Mathematics 4(5). Reston, VA: National Council of Teachers of Mathematics.
Parker, R. E. (1993). Mathematical power: Lessons from a Classroom. Portsmouth,
NH: Heinemann
Powell, A. B. & Frankenstein, M. (1997). ). Ethnomathematics: Challenging
eurocentrism in mathematics education. Albany, NY: State University of New
York Press.
Van de Walle, J. (1997). Elementary and Middle School Mathematics (3rd Edition).
New York: Longman.
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