Facilitating Meaningful Mathematics Discussions

Matt Wallace STEM Academy

Goal for this Morning…

Logistics• Cell phones? Answer them… outside please! • Questions? Ask them… at any time! • Today’s handouts

Overview of the Morning✓ Logistics • The point of mathematics discussions • Solve and discuss the “Half of a Whole” task • Watch a video clip of a 3rd grade teacher facilitating a small group

discussion of this task • Discuss ways the teacher supported students’ learning of mathematics • Connect specific teacher actions to Effective Mathematics Teaching

Practices • Questions & Resources

Essential Questions• What is the point of discussion in mathematics? What does an

meaningful mathematics discussion look like and why is i effective? What work is involved in leading a class discussion?

What fraction of each rectangles is

shaded?

Classroom Discussions

“An excellent mathematics program requires effective teaching that engages students in meaningful learning through individual and collaborative experiences that promote their ability to make sense of and communicate mathematical ideas”

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 7)

Classroom Discussions

Classroom Discussions“In a whole class discussion, the teacher is in charge of the class… However, the teacher is not primarily engaged in delivering information or quizzing. Rather, s/he is attempting to get students to share their thinking, explain the steps in their reasoning, and build on one another’s contributions…The teacher facilitates and guides quite actively, but does not focus on providing answers directly. Instead the focus is on the students’ thinking.” -Ball

“Students who learn to articulate and justify their own mathematical ideas, reason through their own and others mathematical explanations, and provide a rationale for their answers develop a deep understanding that is critical to their future success in mathematics related fields” -Carpenter, Franke, & Levi

Classroom Discussions

“If you would like students to understand, then be sure they are reflecting on what they are doing and communicating about it to others…communicating means talking and listening. It means sharing your method you developed to solve a problem and responding to questions about your method. It means listening to others share this methods and asking questions to make sure you understand.” -Carpenter, et al.

Classroom Discussions

“Encouraging pupils to talk about their conceptions and justify their own strategies of exploration or proof represents a shift in the classroom from a ‘teacher centered’ to more ‘pupil centered’ approach” -Hoyles

Classroom Discussions

What is the point of discussions in mathematics?• Engage students in mathematical reasoning and debate • Do less “telling” and more “eliciting” • Developing student understanding of worthwhile

mathematics (i.e., not a “show and tell”) • Promote constructive critique of ideas while seeking out

efficient mathematical solutions, which not only is a skill that’s transferable to other areas of students lives, it develops the mathematical language skills needed for productive discourse and learning

• Shifting the classroom from a teacher- to student-centered approach

Half of a Whole Task

Half of a Whole Task

Discuss the following questions with your neighbor: 1.What do students need to know and be able to do in order to engage with the task?

2.What mathematical ideas does the task provide an opportunity for students to learn?

Effective Teaching Practice: Establishing mathematical goals to focus learning

“Formulating clear, explicit learning goals sets the stage for everything else.” - Hiebert, Morris, Berk, & Janssen

Learning Goals should:

• clearly state what it is students are to learn and understand about mathematics as the result of instruction

• be situated within learning progressions

• frame the decisions teachers make during a lesson

Half of a Whole Task — Ms. Brooks ClassMs. Brooks Mathematics Learning Goals: Students will understand that…

• A fraction describes the division of a whole region or area into equal parts.

• A fraction is relative to the size of the whole unit.

• If the numerator is half the quantity in the denominator then the fraction is equal to a half.

• Fractions have an infinite number of equivalent forms, regardless of whether or not the pieces of the whole unit are adjacent.

Half of a Whole Task — Ms. Brooks ClassCCSS Addressed:

NumberandOpera-ons–Frac-ons(NF)

Developunderstandingoffrac-onsasnumbers.

3.NF.A.1 Understandafrac2on1/basthequan2tyformedby1partwhenawholeispar22onedintobequalparts;understandafrac2ona/basthequan2tyformedbyapartsofsize1/b.

3.NF.A.3 Explainequivalenceoffrac2onsinspecialcases,andcomparefrac2onsbyreasoningabouttheirsize.(a)Understandtwofrac2onsasequivalent(equal)iftheyarethesamesize,orthesamepointonanumberline.(b)Recognizeandgeneratesimpleequivalentfrac2ons(e.g.,1/2=2/4,4/6=2/3).Explainwhythefrac2onsareequivalent,e.g.,byusingavisualfrac2onmodel.

Half of a Whole Task — Ms. Brooks ClassMath Practices Addressed:

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Half of a Whole Task — Ms. Brooks ClassTeacher: Millie Brooks Grade: 3 This lesson takes place in February. Students have just begun their study of fractions. After reading the instructions aloud, Ms. Brooks asked students to work on the task in small groups. The clip shows the interactions that Ms. Brooks had with one of the small groups, in which there was a disagreement about Figure (d).

Half of a Whole Task — Ms. Brooks ClassAs you watch the video clip, make note of:

• Ways Ms. Brooks supports her students’ learning, and

• The mathematical ideas students are grappling with as they discuss, explain, and try to justify their reasoning.

Half of a Whole Task — Ms. Brooks Class

Share your observations with neighbors

As you watch the video clip, make note of:

• Ways Ms. Brooks supports her students’ learning, and

• The mathematical ideas students are grappling with as they discuss, explain, and try to justify their reasoning.

Effective Mathematics Teaching Practices1. Establish mathematics goals to focus learning. 2. Implement tasks that promote reasoning and problem solving. 3. Use and connect mathematical representations. 4. Facilitate meaningful mathematical discourse. 5. Pose purposeful questions. 6. Build procedural fluency from conceptual understanding. 7. Support productive struggle in learning mathematics. 8. Elicit and use evidence of student thinking.

Effective Mathematics Teaching Practices1. Establish mathematics goals to focus learning. 2. Implement tasks that promote reasoning and problem solving. 3. Use and connect mathematical representations. 4. Facilitate meaningful mathematical discourse. 5. Pose purposeful questions. 6. Build procedural fluency from conceptual understanding. 7. Support productive struggle in learning mathematics. 8. Elicit and use evidence of student thinking.

Discuss with a neighbor… • What were the mathematics teaching practices that you noticed in this

video? • What was the teacher doing that led you to identify this teaching

practice?

Effective Mathematics Teaching Practices1. Establish mathematics goals to focus learning. 2. Implement tasks that promote reasoning and problem solving. 3. Use and connect mathematical representations. 4. Facilitate meaningful mathematical discourse. 5. Pose purposeful questions.6. Build procedural fluency from conceptual understanding. 7. Support productive struggle in learning mathematics. 8. Elicit and use evidence of student thinking.

Facilitating Meaningful Mathematical Discourse

“Discussions that focus on cognitively challenging mathematical tasks, namely those that promote thinking, reasoning, and problem solving, are a primary mechanism for promoting conceptual understanding of mathematics.” Smith, Hughes, Engle, & Stein

Mathematical discourse should: • Build on and honor students’ thinking • Provide students with the opportunity to share ideas,

clarify understandings, and develop convincing arguments

• Advance the math learning of the whole class

Pose Purposeful Questions“Teachers’ questions are crucial in helping students make connections and learn important mathematics concepts. Teachers need to know how students typically think about particular concepts, how to determine what a particular student or group of students thinks about those ideas, and how to help students deepen their understanding.” Weiss & Pasley

Effective Questions should:

• Reveal students’ current understandings

• Encourage students to explain, elaborate, or clarify their thinking

• Make the mathematics more visible and accessible for student examination and discussion

Half of a Whole Task — Ms. Brooks ClassAs you watch the video clip this time, make note of:

• The ways in which Ms. Brooks facilitated meaningful mathematical discourse

• The questions asked by Ms. Brooks.

Be prepared to give examples and to cite line numbers from the transcript to support your observations.

Half of a Whole Task — Ms. Brooks Class

Share your observations with neighbors

As you watch the video clip this time, make note of: • The ways in which Ms. Brooks facilitated meaningful

mathematical discourse • The questions asked by Ms. Brooks.

What work is involved in leading a class discussion?1. Selection of discussion worthy task/problem

2. Launching the discussion

3. Orchestrating the discussion

4. Concluding the discussion

What work is involved in leading a class discussion?1. Selection of discussion worthy task/problem

• Open-ended problems

‣ give two different numbers whose sum is 10

‣ Write three different fractions that simplify (“reduce”) to 2/3.

What work is involved in leading a class discussion?1. Selection of discussion worthy task/problem

• Open-ended problems

• Problems that can be solved in multiple ways

A 4th grade class needs 5 leaves each day to feed its 2

caterpillars. How many leaves would they need each

day for 12 caterpillars?

What work is involved in leading a class discussion?1. Selection of discussion worthy task/problem

• Open-ended problems

• Problems that can be solved in multiple ways

• Problems that reveal misconceptions

What fraction of each rectangles is

shaded?

What work is involved in leading a class discussion?1. Selection of discussion worthy task/problem

• Open-ended problems

• Problems that can be solved in multiple ways

• Problems that reveal misconceptions

• Debatable “either-or” problems Is it possible to make a

triangle with three obtuse angles?

What work is involved in leading a class discussion?1. Selection of discussion worthy task/problem

2.Launching the discussion

• Initial eliciting of S thinking

What work is involved in leading a class discussion?1. Selection of discussion worthy task/problem

2. Launching the discussion

3. Orchestrating the discussion • PROBING S to figure out what s/he means or is thinking, to check if answers

are supported by understanding, to understand wrong answer • FOCUSING Ss to listen and respond • Supporting Ss to make CONNECTIONS • Guiding Ss REASONING • EXTENDING Ss thinking

What work is involved in leading a class discussion?1. Selection of discussion worthy task/problem

2. Launching the discussion

3. Orchestrating the discussion

4. Concluding the discussion

As you reflect on the Effective Discussion Practices examined in this session,

identify one or two ideas or insights that you might apply to your own classroom instruction.

Some Helpful Resources“With my ears to the ground, listening to my students, my eyes are focused on the mathematical horizon” -Ball

‣ Stein, M. & Smith, P., 5 Practices for Orchestrating Productive Mathematics Discussions

‣ Kazemi, E. & Hintz, A. Intentional Talk: How to structure and lead productive mathematical discussions

Thank You!

Please contact me for these slides or any other information.

Matt Wallace

mtwallace@ucdavis.edu