17 October 2001

Purpose

The purpose of the today's session is to explore the communication of mathematical thinking. In addition, we will think about appropriate response to student work.

Process

We will explore two sets of problems centered around algebraic and/or geometric content.

There is one ground rule: stay focused on the question at hand; please, no reading ahead. This will allow us to focus onprocesstogether.

We will use the same protocol for each problem.

- Working
alone, each person is to respond to the problemin writing.

- Do NOT identify yourself on your paper.

- Your solution should have a short statement of the "Answer".

- The "Answer" should be followed by an explanation that allows the reader to know what you were thinking. The explanation should be detailed enough to clearly communicate that you understood the essential relationships and processes, even if there are arithmetic errors.

- The papers will be collected.

- In small working groups of two or three, identify the essential elements of a correct solution to the problem. We will seek whole group consensus on these essential elements.

- Our own papers will be distributed among our group, and we will critique our own work. We will practice giving constructive or helpful feedback as is appropriate.

- We will look at highlighted (correct) student solutions.

What different approaches or ways of communicating mathematical content and processes are evident?

- For some problems, we will look at a selection of incorrect student responses.

Based on actual or anticipated errors, what misunderstanding(s) do you think students might have in trying to solve this problem?

What intervention can you suggest to help the student if she or he were here with us now?

Geometry-Based Problems

- Divided Rectangle

- Descartes' Triangle

- Framed!

- Rotating a Triangle

- Triangles and String

- Picking Painted Cubes

- Walking Fast

- A Triangle/Square Ratio

- As Far as the Eye Can See

Algebra-Based Problems

- Phone-O-Graph Phollies

- Great One

- Jack and Jill Go Jogging

- At Least This Many

- "Ace" Trotter

- World's Meanest Math Teachers

- A Dog, a Hog and a Frog

- Painting Partners

- An "Average" Problem

- Bugs Bunny's Money

Thinking about the HSPA

- How is HSPA different from HSPT?

- What should you and your students know about how the test is scored?

- What can you be doing in your classrooms to help students perform as good as they can?

What more could we do to be supportive?

- site visits and in-class support by Anita or Shelly?

- possible topice for future in-service opportunities?

- other ideas?

Evaluation Form

Connecting with the New Jersey Mathematical Framework:

Content Clusters

- Number Sense, Concepts, & Applications
- Spatial Sense & Geometry
- Data Analysis, Probability, Statistics, & Discrete Mathematics
- Patterns, Functions & Algebra

Cross-Cluster Themes

- Problem Solving
- Communication
- Connections
- Reasoning

- Tools & Technology
- Numerical Operations
- Measurement
- Estimation
- Excellence & Equity

Recurring Content Strands

- Discrete Mathematics
- Building Blocks of Calculus