1

Math CAMPPP 2012Breakout 4A

Questioning, Listening and Responding toPromote Students’ Mathematical Thinking

Learning Goals

We will:

• use questioning and listening strategies that evoke and expose thinking

• explore the concept of fraction as a quotient (partitive and quotative)

22

View with a Purpose

http://cooperativelearning.nuvvo.com/lesson/9592-seinfeld-teaches-history

Grade 8 math class in the OCDSB

3

View & Discuss

• What are the key differences between these two classes?

4

Individual Reflection• Questioning resources• Peruse a resource of your choice

Consider how they can be used to help kids and ourselves develop questioning techniques.

Quotient – PartitiveBrownie Problem

6

Sue baked brownies.Her son got in and ate some. There were only 7 left. Sue’s daughter and her friends walked in. They decided to split them up.

How much will each of the 4 girls get?

Show your thinking.

Questioning Simulation

• solve the problem

(anticipate student thinking)• identify potential questions • what might the learning goals be

What are some of the challenges in questioning?• Making sense of another’s thinking• Changing question in the moment

Representations & Learnings

• Fair sharing foundational to division• Informal division of whole numbers with

fraction remainder• Area model (represents a number)• Set model (3 pieces, 3 brownies)• Renaming of fractions (1¾=7/4=1+½+¼)

Student responses

• Provide insight into their thinking• Illuminate ways the students understand

concepts• Provide examples of ways of thinking for

other students

10

Part 2 …

• “What is the shortest word in the English language that contains the letters: abcdef?

12

Answer: feedback.

Standing Conversation

It is worth noting, right from the start, that assessment is a human process, conducted by and with human beings, and subject inevitably to the frailties of human judgement. However crisp and objective we might try to make it and however neatly quantifiable may be our ‘results’, assessment is closer to an art than a science. It is, after all, an exercise in human communications.

Growing Success, p. 29

1313

Minds On

Standing Conversations

Feedback provides students with a description of their learning. The purpose of providing feedback is to reduce the gap between a student’s current level of knowledge and skills and the learning goals.

Growing Success, p. 34

1414

Minds On

Standing Conversations

Ongoing descriptive feedback linked specifically to the learning goals and success criteria is a powerful tool for improving student learning and is fundamental to building a culture of learning within the classroom.

Growing Success, p. 34

1515

Minds On

Roles …

1 – Teacher

2 – Observers

and Students

Ribbon Problem

• You have 7 yards of ribbon. It takes 3/4 of a yard to wrap a gift. How many gifts can you wrap?

Responding by Listening Simulation

• Teachers with a facilitator• Observers with a facilitator• Students work on the problem

A framework for examining questions

• Checking for understanding• Promoting group interaction• Teacher reviewing or summarizing work• Asking students to explain their thinking• Looking for alternate ways• Prompting to go farther• (Facilitating teacher observation)

Questioning to …

• make observations that helps improve understanding

• fosters student sense of self-efficacy • playback to student what I see in their work • get kids to reflect on their work and see where they

can go next • probe student thinking • develop a student-to-student network of dialogue

20

Summary …

• our questioning can serve to elicit and promote mathematical thinking

• a fraction can be a verb – the action of quotient (division) with two different but related meanings

21

Fraction as a Quotient (partitive)

• A fraction can represent division• e.g. 7 brownies divided among 4 people• Number of groups is known (4)• The amount per group is unknown

22

Fraction as a Quotient (measurement - quotative)• You can divide by a fraction• e.g. 7 yards of ribbon, ¾ of a yard to wrap a

gift• Number of groups (gifts) is unknown • The amount per group (gift) is unknown (¾)

23