Workshop
5 -- Idea-Making
Download Workshop 5 in
PDF
This workshop
will focus on student idea-making in mathematics. Constance Kamii
will explain how you can adapt your teaching to help students construct
their own mathematical ideas. You will see video of students engaged
in "mind mathematics" articulate and defend their strategies
to classmates, and you will consider the value of using games to facilitate
mathematics teaching and learning.
Constance
Kamii
Professor of
Early Childhood Education at the University of Alabama at Birmingham,
Constance Kamii studied under Jean Piaget for a dozen years, first
as a postdoctoral research fellow and later as an adjunct professor
at the University of Geneva. She developed a preschool curriculum
based on Piaget's theory, especially in science, mathematics, and
the sociomoral realm, and is now developing an elementary math program
based on his theory. Kamii is the author of Young Children Reinvent
Arithmetic; Young Children Continue to Reinvent Arithmetic, 2nd Grade;
and Young Children Continue to Reinvent Arithmetic, 3rd Grade.
Workshop 5 Timeline
Getting Ready
-- 30 Minutes
15 minutes--Game
Play
You were asked
to bring in some common household board, card, and dice games. Divide
into small groups and select a game to play. While you are playing,
think about what (if any) math skills students would need to play
the game successfully. What are some math concepts (if any) that students
might learn from playing the game?
15 minutes--Game
Discussion
Rejoin the large
group and discuss the following: How does playing games compare to
more traditional math exercises such as worksheets, flash cards, or
problem sets? What are the advantages of games? Disadvantages?
Watch the Workshop
Video -- 60 Minutes
Going Further
-- 30 Minutes
30 minutes--Tricks
and Procedures
When asked in
her interview what headline she would give to a newspaper article
about her approach to learning, Dr. Kamii said, "Traditional
math education harms children's development of numerical thinking."
She went on to explain her belief that teaching children to memorize
rules and algorithms such as carrying, borrowing, and long division
prevents them from inventing their own solutions to problems and forces
them to give up their own thinking.
Do you agree
that teaching algorithms and procedures, tricks and equations, has
a negative effect on student learning? What are the advantages of
algorithms from a teacher's perspective? From a learner's perspective?
What are the disadvantages? How would your school district react to
Dr. Kamii's statement?
For Next Time
Homework Assignment
How comfortable
are you with letting students develop and pursue their own inquiry?
To test your comfort level, try this -- during the next week, incorporate
into a lesson some sort of class discussion in which students can
talk about their opinions or rationales for solving something. During
this class discussion, see how many times you can let the comments
pass from student to student without an intervening question or comment
from you. It's not easy!
How many students
were able to speak consecutively before you spoke? When did you intervene?
Why? What sort of discussion was happening when you jumped in? What
could you do next time to let the discussion go further on its own?
Reading Assignment
In preparation
for Workshop 6, please read "Developing the Spectrum of Human
Intelligence"by Howard Gardner. (All readings are included in
the Appendix.)
Moon Journal
You might want
to reflect on the following in your Moon Journal:
- Why does the Moon appear to shine?
- Why does the Moon appear to change its shape?
- Is there an order to the Moon's phases? If so, can you determine
the order?
Suggested Activity
Modeling
the Phases of the Moon
To do this activity
effectively, the room must be as dark as possible. Darken the room
by closing the blinds and covering all window and door cracks with
black paper or cloth and tape.
Materials:
Lamp, Extension cord, Clear light bulb (75 watts or more), 3-inch
Styrofoam ball, Craft stick
Instructions
- Make a "handle"
for your Styrofoam ball by carefully pushing a craft stick into
the ball. Hold the handle so the Moon ball is positioned upright.
- You will
be part of a model that portrays the phases of the Moon. In the
model, your head will represent the Earth, the Styrofoam ball will
represent the Moon, and the lamp will represent the Sun.
- Place the
lamp in the center of the room and turn it on. Turn off the room
lights and then stand approximately two arm-lengths away from the
lamp.
- Hold the
Moon ball directly in front of the lamp and at arm's length from
your body, pointing upwards approximately 45 degrees. Notice that
as you hold the Moon ball in front of your body and turn around,
sometimes part of the ball is lit, and sometimes the whole hemisphere
facing you is lit.
- Keep turning
until you can see a thin crescent lit up on the Moon ball.
- Continue
moving in the same direction until the Moon ball looks like a half-lit
circle.
- Continue
moving in the same direction until the ball looks completely lit.
Questions
- Is the brightest
side of the Moon facing towards or away from the Sun?
- For the Moon
to appear "fuller," how does it have to change its position
relative to the Sun?
- When the
Moon is full, is it on the side of the Earth that's closest to the
Sun, or the side that's farthest away from the Sun?
- Are the phases
of the Moon the same in the northern and southern hemispheres?
Extensions
On a sunny day
when the Moon is visible, go outside with your Styrofoam Moon ball.
Stand facing the Moon, holding out your Moon ball at arm's length
"covering" the Moon in the sky. The Sun will shine on the
ball and illuminate it exactly as it illuminates the Moon.
Adapted from:
Foster, G.W. (1996). Look to the Moon. Science
and children. 34(3), 30-33.
Braile, S. (1994). Moon phase modeling. In N.B. Ball,
H.P. Coyle, & I.I. Shapiro (eds.), Project SPICA. Kendall/Hunt
Publishing Co: Dubuque, Iowa.
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