Transcript How do teachers and students “leverage” the mathematics itself?

How do teachers and students
“leverage” the mathematics
itself?
TDG Seminar Plenary
Feb 14, 2009
J. Michael Shaughnessy
Portland State University
[email protected]
Valentine’s thought:
Let’s face it, We’re all in this profession
because of two loves:
– The Math
– The Kids
and
What trips the switch, what get’s
our creative Math juices flowing?
What helps our students to
“move” to a new place in their
thinking when we are DOING
Mathematics?
How do we “leverage” our
mathematical explorations?
What leverages creativity in other
areas?--I looked for analogies,
for the ‘fuel’ of other disciplines
Environments
• Music
• Literature
• Mathematics
Music
Environments
Tools --> created object
• Music
• Notes --> Melodies
• Literature
• Mathematics
Here is a musical example from
Ernesto Nazareth
From notes,
to melody,
to imbedded theme.
Literature
Environment
• Music
• Literature
• Mathematics
• Tools --> created
object
• Notes --> Melodies
• WordsPoetry/Essays,
images
An example from literature,
words to poetic images:
What one inevitably sees along
river banks is:
River, rocks, water, dirt,-->
“Sediments flowing in from the west formed deltas,
floodplains, and tidal flats, which indurated into
these fine-grained sedimentary rocks thinly laid
deposits of a restful sea, lined with shadows as
precise as the staves of a musical score, ribboned
layers, an elegant alteration of quiet siltings and
delicious lappings, crinkled water compressed,
solidified, lithified. A loose fragment of shale the
size of a quarter starts a stream of little stony
pumpkin seed chips with a quiet rustling and
whispering.”
From Tanner Trail and Mesquite Thickets by Ann
Haymond Zwinger
Mathematics
Environment
Tools --> created object
• Music
• Notes --> Melodies
• Literature
• Words-Poetry/Essays,
images
• ????
• Mathematics
???? =
REPRESENTATIONS
of concepts and processes
They can mediate, or even reveal,
the ‘big ideas’ in a mathematical
problem
Representation?
What do dictionaries say…
• 1. That which represents another.
• 2. A Figure, image or idea that
substitutes reality.
• 3. A Theatrical performance.
“Representation is the map that
connects the source concept with
a target object.”
• Ex: Real numbers <---> Number line
It is the correspondence that is the representation,
not the real numbers, not the line itself, but rather
the map between the two.
---The Role of Representation in School Math
2001 NCTM Yearbook
Representations can be
• Internal-- in our own mind, personal
• External-- easy to share and communicate
• Invented-- created by students and teachers
• Presented--conventionally accepted, put forth by
teachers as ‘given’
The Roles of Representation in School Mathematics,
2001 Yearbook of NCTM (Al Cuoco, Editor)
Let’s play with some
mathematical tasks, and as we do
so, pay attention to when our
representations of the tasks are
moving us to a new place in our
thinking
Ratio of Regions in a Square
A
B
IV
E
III
I
II
D
C
Ratio of Regions in a Square
E is the midpoint of the side AD. What is
the ratio of the areas of the four regions
I, II, III and IV to one another?
Work on it, keep track of how you are
using representations in your process…
Girl children are especially
prized in a culture of one of the
Pacific islands. On the other
hand, resources are limited on the
island. So, the elders institute the
following policy:
One Girl Families
Child capable parents may have
babies until they have a girl child.
Then they must stop.
What will such a policy do to the
population composition on the
island in the long run?
These are called “reversie pairs.”
68
x 43
86
x 34
21
x 84
12
x 48
36
x 42
63
x 24
What do you notice?
What do you wonder about?
• What makes this “work?”
The Representation Standard in PSSM
Instructional programs from Pre K – Grade 12
should enable all students to—
• Create and use representations to organize, record,
and communicate mathematical ideas
• Select, apply, and translate among mathematical
representations to solve problems
• Use representations to model and interpret
physical, social, and mathematical phenomena
Excerpts from the Representation
Standards
• A major responsibility of teachers is to create a
learning environment in which students’ use of
multiple representations is encouraged (PreK -2)
• Each representation reveals a different way of
thinking about the problem. Giving attention to the
different methods will help students see the power
of viewing a problem from different perspectives.
(3-5)
• Representations can help students to model
relatively large scale, significant problems. 6-8)
• A major responsibility of teachers is to create a
learning environment in which students’ use of
multiple representations is encouraged (Prek-2)
• Part of a teacher’s role is to help students connect
their personal images to more conventional
representations. (9-12)
It seems to me that representations
provide the creative fuel for inventing,
and doing, mathematics, just as words do
in poetry, or, notes in music. It’s how the
words, the notes, the representations are
put together that can move us to a new
place, to a poem, or to a song, or to a
solution for a mathematical problem.
Representations can capture the big
mathematical ideas, can gives us the
power to leverage (MOVE)
mathematical thinking.
Stay in Love with Your Math!
And with your Students!
And, keep both of them on the
MOVE!
Happy Valentines Day