e-Learning for Educators
Course Information

Geometric Measurement

Course Description

In this course, participants will become familiar with the Van Hiele framework and how it can be used to understand the challenges of teaching particular geometric measurement tasks for students in the middle-grades.  Participants will learn about and discuss ways to use the framework as a basis for teaching perimeter, area, surface area and volume to ensure greater student success.  In addition, participants will learn how to use a protocol for examining student work in order to reflect on and analyze students’ thinking to help them plan the next steps in their teaching process.  For the final project, participants will use the Van Hiele framework to develop or modify a classroom task focused on geometric measurement.

Participants in this course must be currently employed as a middle grades mathematics teacher or school administrator.

Course Syllabus

### Catalog Description

In this course, participants will become familiar with the Van Hiele framework and how it can be used to understand the challenges of teaching particular geometric measurement tasks for students in the middles grades.  Participants will learn about and discuss ways to use the framework as a basis for teaching perimeter, area, surface area and volume to ensure greater student success.  In addition, participants will learn how to use a protocol for examining student work in order to reflect on and analyze students’ thinking to help them plan the next steps in their teaching process.  For the final project, participants will use the Van Hiele framework to develop or modify a classroom task focused on geometric measurement.

### Prerequisites

This workshop is designed for teachers, curriculum specialists, professional development specialists, or other school personnel. Participants are expected to have regular access to computers, and proficiency with email and current web-browsers.

### Goals

Participants will:

• read and discuss relevant research in teaching geometric measurement in the middle grades, particularly develop familiarity with levels of geometric reasoning,
• learn how to analyze student work to inform instruction,
• learn approaches to teaching geometric measurement that emphasize both conceptual and procedural knowledge, particularly in linear measurement, area, surface area, and perimeter,
• explore the relationships between perimeter and area and between area and surface area, and
• explore visualization, spatial reasoning, and geometric modeling as vehicles to build mathematical skills;bBecome familiar with virtual manipulatives and hands on activities designed to promote both conceptual and procedural knowledge around geometric measurement.

### Assessment and Course Requirements

This workshop is divided into six one-week sessions, each of which includes readings, an activity, and an online discussion among workshop participants. The estimated time for completing each session is approximately 6.5-7 hours.  Participants are expected to be engaged in course activities for approximately 45 hours during the six-session course.

### Course Products

The final project provides practice in 1) recognizing the demands of a task and determining what parts will be challenging for students, 2) applying a student work protocol to identify student understandings and 3) preparing an instructional plan using the information from the student work protocol. To complete the final project teachers will reflect on the videos viewed in some sessions and consider concepts learned in the workshop. Teachers will then select a student work sample and apply a student work protocol in order to look for evidence of what the student understands and does not understand. Teachers will then write a reflection about where the student is in terms of concept development and understanding and prepare an instructional plan of action.

### Discussion Participation

Participants will be evaluated on the frequency and quality of their discussion board participation.  Participants are required to post a minimum of two substantial postings each session, including one that begins a new thread and one that responds to an existing thread. Postings that begin new threads will be reviewed based on their relevance, demonstrated understanding of course concepts, examples cited, and overall quality.  Postings that respond to other participants will be evaluated on relevance, degree to which they extend the discussion, and tone.

### Session One: Understanding Levels of Geometric Reasoning

This session introduces a summary of the levels of geometric reasoning that will be used throughout the course as participants analyze geometric measurement tasks and corresponding student work.  Through these examinations, teachers will be able to alter and improve their instruction to promote both procedural and conceptual understanding of geometric measurement.  Participants will do some math activities that illustrate the levels of geometric reasoning and consider how the framework can be applied to decisions about classroom tasks and instruction.  In addition, participants will start thinking about how using different types of models – both virtual and “real life” – can help in student understanding.

### Session Two: Developing a Protocol for Examining Student Work

In this session, participants will examine the topic of linear measurement as foundational to learning common geometric measurement topics including perimeter, area and surface area.  Through the use of a Looking at Student Work Protocol that is aligned to the common framework, participants will experience a process for examining student work to assess understanding and inform instruction. Participants will determine what students find challenging in learning how to measure and interpret measurements.  Participants will explore strategies and activities for teaching linear measurement that employ both standard and non-standard units of measure and are designed to build conceptual understanding.

### Session Three: Understanding Area, Surface Area and the Role of Spatial Reasoning

This session examines two-dimensional geometric measurement on two-dimensional figures (area) and on three-dimensional figures (surface area).  Through readings and math activities, participants explore two conceptual issues that are often at the root of student misconceptions (1) why square units are used to measure area, and (2) the need for solid spatial reasoning skills to successfully complete both area and surface area problems.  Participants will watch video of students completing similar math activities and will use the videos and corresponding student work to analyze student understanding on the same problems using the student work protocol introduced in the prior session.

This is the first of several instances in the course where participants combine their growing understanding of levels of reasoning with the protocol for analyzing student work to deepen their understanding of ways to select geometric measurement tasks appropriate to their student readiness.

### Session Four: Perimeter and Area Relationships

This session focuses on the relationship of perimeter and area.  By examining these two geometric measures together, participants explore ways to build strong conceptual understanding of each.  Participants will complete various math activities using online manipulatives and hands on activities designed to reach a variety of learners.  Participants will then evaluate the demands of these activities based on the common framework and consider how the tasks match their students’ skills and abilities.  Throughout, participants will be asked to consider ways to support students in moving toward a deeper conceptual understanding in completing these tasks.

### Session Five: Why Not Just Use a Formula?

This session examines ways to introduce, understand, and use formulas for geometric measurement so that students develop meaning for the use of the formulas. Readings in this session underscore the difficulty students have when simply memorizing or applying formulas with no conceptual understanding and make the case for teaching formulas for geometric measurement with understanding through the use of manipulatives, models and patterns.  Participants will explore the use of manipulatives to examine patterns and make generalizations about sets of geometric measurements that lead to the development of formulas – a process that parallels work that students have often done with patterns in algebraic thinking.  Through the use of classroom video and student work, participants will evaluate a surface area and volume problem using the common framework and will analyze student understanding using the student work protocol.

### Session Six: Putting It All Together

In this session, participants will demonstrate their learning in the course through discussion, reflection, and analysis.  Participants will complete a final project in which they synthesize the prior experiences of the course and apply their skills to the analysis of common assessment tasks.  Drawing upon research about teaching and learning geometric reasoning and through the examination of tasks and student work, participants will make instructional decisions to move students toward understanding.