In this course, participants will learn about different components and levels of proportional reasoning. They will complete classroom tasks that illustrate ways to help their students develop stronger conceptual understanding about the components of proportional reasoning They will explore video and written examples of student work and will conduct a student interview in order to reflect on how students approach proportional reasoning and to learn how to identify evidence of proportional reasoning among students. For the final project, participants will create a lesson that targets a component of proportional reasoning and that incorporates questioning techniques learned in the course. They will provide a rationale for the lesson that relates to the material learned in the course and reflect on their own learning.
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.
This course will enable participants to:
Understand and identify different components of and levels of proportional reasoning;
Broaden their repertoire of classroom tasks to use to develop students’ conceptual understanding of proportionality;
Develop skills and tools for analyzing students’ thinking about proportionality;
Apply those skills and tools to a lesson on proportional reasoning that is appropriate to middle school students.
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 time for completing each session is estimated to be approximately five hours. Assessment will include participation in course discussions and completion of assignments and a culminating classroom-focused project. Participants are expected to be engaged in reading, reflection and course activities for approximately 30 hours during the six-session course.
Participants will administer the Proportional Reasoning Pre-Assessment to 2-3 middle school students and review the results. From those results, they identify a component of proportional reasoning with which their students need further work. Then they will complete a Personal Reflection and answer a question related to this work. Participants will also conduct a student interview and submit another Personal Reflection on the data gathered in the interview. Finally, participants will create a lesson that targets a component of proportional reasoning identified from the pre-assessment as an area of weakness.
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: Recognizing When a Situation is Proportional
In this session, participants will become familiar with one component of proportional reasoning: determining when a situation is proportional. In order to understand challenges students may have in identifying situations that call for proportional reasoning, participants will complete problems that highlight different approaches to helping students think about the question of when it is appropriate to use proportional reasoning in a problem. In addition, participants will view a video clip of students discussing a problem, and they will analyze student dialogue about proportionality problems to determine students’ ability to apply proportional reasoning appropriately. In preparation for the final project they will select 2-3 students to complete the pre-assessment on proportional reasoning. An alternate plan is provided for participants who do not have students available.
Session Two: Understanding Ratios
Participants will become familiar with two additional components of proportional reasoning: 1) understanding the invariance of ratio and 2) seeing ratios as composite units or “units of units.” They will complete sample problems that illustrate this component of proportional reasoning and analyze student work on the same problems as preparation for the discussion in this session. Additionally, they will complete a Pattern Block Ratios activity, a visual activity that uses pattern blocks to make equivalent “shape-fractions.” During this session, participants are learning to recognize and identify examples of evidence that students are developing an understanding of the two components of proportional reasoning introduced in the session readings. In preparation for the final project, participants will analyze samples of student work.
Session Three: Reasoning about Proportionality without Calculations
This session enables participants to become familiar with the concept of reasoning proportionally without calculations. Participants will complete problems in the Reasoning without Calculations activity, which asks them to use proportional reasoning to solve problems by focusing on the relationships rather than specific values.
They will view a video of students discussing their solutions to these kinds of problems and analyze students’ dialogue to identify evidence of use of this component of proportional reasoning. Additionally, participants will review student work to identify areas of student strength and weakness in proportional reasoning. After this work, they will complete one section on a Personal Reflection page.
Session Four: Using Interviews as a Window into Students’ Thinking
In this session, participants will become familiar with some of the ways that students reason about proportional situations in common proportion problems. They will view three video clips of students solving different proportional reasoning problems and identify different student misconceptions related to proportional reasoning as well as possible sources of those misconceptions. They will also identify instances of students providing a correct answer, but still not understanding the underlying mathematics. Participants will learn about the purpose for and method of conducting student interviews. They will conduct a student interview as a tool for analyzing students’ proportional reasoning on a problem. An alternative activity is provided for participants who do not have access to students for this activity. The discussion relates to the kinds of questions teachers ask to resolve student misconceptions.
Session Five: Questioning Strategies for Classroom Discussions
During this session, participants will learn about questioning techniques to draw out and illuminate students’ thinking. They complete a student interview and analyze the results for evidence of proportional reasoning. Alternate ideas are provided for participants who do not have access to students of their own. Participants will begin to develop a lesson plan that applies concepts of proportional reasoning addressed in this course. They will also complete an additional section on the Personal Reflection page begun in session three of this workshop.
Session Six: Building on Students’ Proportional Reasoning in the Classroom
Participants will read a brief case that describes students’ responses to a lesson on cross-multiplication and poses a dilemma faced by the teacher in the case regarding how to proceed with those students. Participants will use the discussion to propose solutions to the teacher's dilemma in this case. Participants will complete their lesson that targets a component of proportional reasoning identified from the students pre-assessment as an area of weakness. They will incorporate questioning techniques into the proposed lesson and provide a rationale for the lesson that relates to the material learned in the course. Additionally, participants will reflect on their own learning by comparing their prior definition of proportional reasoning to their current one.