Math Fellows

January 22, 2014

0. Setting Goals and Selecting Tasks

1. Anticipating

2. Monitoring

3. Selecting

4. Sequencing

5. Connecting

The Five Practices (+)

Agenda Do some math! Review the five practices and apply to a

new task Reflect on implementation of practices Using questioning to support the

enactment of the practices/ or in efforts to engage students within discussion

01: Setting Goals

Without explicit learning goals, it is difficult to know: What counts as evidence of student learning How students’ learning can be linked to particular

instructional activities How to revise instruction to facilitate particular

instructional activities How to revise instruction to facilitate students’ learning

more effectively

“Formulating clear, explicit learning goals sets the stage for everything else.”

Hiebert

Our Task: The Sum of Two Odd Numbers

Prove the following conjecture:

The sum of any two odd numbers is an even

number.

5

Our Goal for the Lesson

1. Realize that examples are not enough to show that a

claim is always true.

2. Recognize that there are many different ways to

prove that a claim is true and that it is not the form

that matters but, rather, the consideration of all cases

and creation of a clear and logical argument.

3. Understand that there are reasons WHY

mathematics works the way it does that can be

explored and explained.7

Our Task: The Sum of Two Odd Numbers

Prove the following conjecture:

The sum of any two odd numbers is an even

number.

8

Anticipated Solution Methods

Picture – odd numbers represented as a number of sticks, grouped by 2, showing one left over

Sketch of a Rectangle – odd numbers represented as a 2-by-n rectangle with one extra square

Logical Argument Algebra

2: Monitoring Responses

Which responses might you look for in your/and your teams student work?

Connecting

Picture (H) Loner Number (G) Algebra (A) Empirical Example (E)

Connect the “left over” dot in Student H’s response, the “loner number” in Student G’s response, and the +1 in the algebraic response.

Connect the grouping by two in H with the 2x in A

“The five practices can help teachers manage classroom discussions productively. However, they cannot stand alone…

In addition, teachers need to develop a range of ways of interacting with and engaging students as they work on tasks and share their thinking with other students.

This includes: having a repertoire of specific kinds of

questions that can push students’ thinking toward core mathematical ideas

methods for holding students accountable to rigorous, discipline-based norms for communicating their thinking.”

Questioning to Support Enactment of the Practices

Type Description

Exploring mathematical meanings and/or relationships

•Points to underlying mathematical relationships and meaning•Makes links between mathematical ideas and representations

Probing, getting students to explain their thinking

•Asks student to articulate, elaborate, or clarify ideas

Generating discussion •Solicits contributions from other members of the class

“These questions do not take over the thinking for the students by providing too much information or by ‘giving away’ the answer or a quick route to the answer. Rather, they scaffold thinking to enable students to think harder and more deeply about the ideas at hand.” p.62

Exploring Questioning in Regina Quigley’s Classroom

The background: 4th grade classroom Geometry unit Before the lesson, students found areas of

rectangles and squares

Teacher’s goal for lesson: students will construct the formula for finding the area of a right triangle by manipulating premade cardboard right triangles against a backdrop of grid paper.

Regina Quigley’s Lesson

Individually, read the vignette.

With a partner:1. Highlight questions

2. Categorize questions as to typeExploring, Probing, Generating

Questioning to Support Enactment of the Practices

Type Description

Exploring mathematical meanings and/or relationships

•Points to underlying mathematical relationships and meaning•Makes links between mathematical ideas and representations

Probing, getting students to explain their thinking

•Asks student to articulate, elaborate, or clarify ideas

Generating discussion •Solicits contributions from other members of the class

“These questions do not take over the thinking for the students by providing too much information or by ‘giving away’ the answer or a quick route to the answer. Rather, they scaffold thinking to enable students to think harder and more deeply about the ideas at hand.” p.62

Examining the Explore Questions

With a small group Identify Explore questions in the vignette

For each Explore question, consider: What is the purpose of the question? What is the underlying mathematical idea or enduring

understanding?

Whole group share out

Reflection

Individually, consider:

Where does questioning fit within the Five Practices?

Turn and Talk

Reflecting on Implementation in your TutorialIndividually, take a few minutes to write: What have you done to support the five

practices in your class? What was the impact?

Whole group: What new ideas did you get from the discussions?

Implications for your Work:

How can you use today’s learning of strategies to help engage students in mathematical discussion within your tutorials?