How do teachers and students “leverage” the mathematics

itself?

TDG Seminar Plenary

Feb 14, 2009

J. Michael Shaughnessy

Portland State University

Mikesh@pdx.edu

Valentine’s thought:

Let’s face it, We’re all in this profession because of two loves:

– The Math and

– The Kids

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 disciplinesEnvironments

• Music

• Literature

• Mathematics

Music

Environments

• Music

• Literature

• Mathematics

Tools --> created object

• Notes --> Melodies

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

• Words-Poetry/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

• Music

• Literature

• Mathematics

Tools --> created object

• Notes --> Melodies

• Words-Poetry/Essays, images

• ????

???? = 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

E

A B

CD

I

I I

I I I

I V

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 FamiliesChild 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 86 x 43 x 34

21 12 x 84 x 48

36 63 x 42 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