5 Pillars of 5 Pillars of MathematicsMathematics

Training #1: Mathematical Discourse

Dawn PerksGrayling B. Williams

Today’s AgendaToday’s Agenda

• Quick Introductions• Statement of Purpose and Expectations and Session Focus• Which Does Not Belong?• Task 1: “It All Adds Up!”• Debriefing: Accountable Talk• Talk Moves• Task 2: “Eric, The Sheep”• Debriefing: What did we learn?• Task 3: The Expressions• Debriefing• Transforming Mathematical Tasks• Debriefing• Where do we go from here?

Teaching and Assessment Framework

5 Pillars of Mathematics1.Reasoning to make sense of mathematics2.Productive use of discourse when explaining

and justifying mathematical thinking3.Procedural fluency4.Flexible and appropriate use of mathematical

representations5.Confidence and perseverance in solving

1. Reasoning to make sense of mathematics

1. Make sense of problems andpersevere in solving them.7. Look for and make use ofstructure.8. Look for and express regularityin repeated reasoning.

2. Productive use of discourse whenexplaining and justifying mathematicalthinking

3. Construct viable arguments andcritique the reasoning of others

Which does not belong?Which does not belong?Why?Why?

2, 6, 5, 10

Which does not belong?Which does not belong?Why?Why?

9, 16, 25, 43

Which does not belong?Which does not belong?Why?Why?

2, 3, 15, 23

Course GoalsCourse GoalsThis course is designed to help you:• Strengthen your math content and pedagogical

knowledge for the purpose of making math accessible for students;

• Understand how students learn mathematics; and• Implement instructional strategies that promote

thinking, reasoning, and making sense of mathematics as called for in the Common Core State Standards

Task #1Task #1

It All Adds Up!

Why is talk critical to Why is talk critical to teaching and learning?teaching and learning?

Positive Influences of Mathematical Discourse

• Accountable talk can reveal understanding andmisunderstanding.

• Accountable talk supports robust learning by boosting memory.

• Accountable talk supports deeper reasoning.• Accountable talk supports language development.• Accountable talk supports the development of

social skills.

Talk Moves• Revoicing• Repeating• Reasoning• Adding on• Waiting

Video 3.2bClassroom Discussions: Using Math Talk to Help Students Learn, 2009

15 minute break15 minute break

Task #2Task #2Eric The SheepEric The Sheep

It’s a hot summer day, and Eric the Sheep is at the end of a line of sheep waiting to be shorn. There are 50 sheep in front of him. Eric is impatient, and every time the shearer takes a sheep from the front of the line to be shorn, Eric sneaks up two places in line.

What Are Good Tasks?

What Are Good Tasks?What Are Good Tasks?

• They help students make sense of the mathematics.• They are open-ended, whether in answer or approach.• They empower students to unravel their

misconceptions.• They not only require the application of facts and

procedures but encourage students to make connections and generalizations.

• They are accessible to all students in their language and offer an entry point for all students.

• Their answers lead students to wonder more about a topic and to construct new questions as they investigate on their own.

Asking Essential QuestionsAsking Essential Questions

What do we want our students to know and understand about variables?

Standards for Mathematical Standards for Mathematical ContentContent

• 6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

• 6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.

• 6.EE.4 Identify when two expressions are equivalent • 7.EE.4 Use variables to represent quantities in a real- world or

mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

• HSN-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.

• HSA-SSE.A.1 Interpret expressions that represent a quantity in context.

Task #3Task #3

The Expressions

Knowledge:Knowledge:Conceptual vs. ProceduralConceptual vs. Procedural

Consists of well-defined concepts, more informal mathematical ideas, and relationships among ideas, concepts, and skills.

Involves knowledge of facts, symbols, rules, and procedures.

"When concepts and procedures are not connected, students may have a good intuitive feel for mathematics but not solve the problems, or they may generate answers but not understand what they are doing.” James Heibert, author of The Teaching Gap

Rigor

ConceptualUnderstanding Application

Skills and Procedures

“It is possible to have procedural knowledge of a topic and to have little or no conceptual knowledge.

However, without knowledge of the important concepts and ideas, it is impossible to truly understand that topic.”

--Classroom Discussions: Using Math Talk to Help Students Learn, 2009

5 Pillars of Mathematics1.Reasoning to make sense of mathematics2.Productive use of discourse when explaining

and justifying mathematical thinking3.Procedural fluency4.Flexible and appropriate use of mathematical

representations5.Confidence and perseverance in solving

1. Reasoning to make sense of mathematics

1. Make sense of problems andpersevere in solving them.7. Look for and make use ofstructure.8. Look for and express regularityin repeated reasoning.

2. Productive use of discourse whenexplaining and justifying mathematicalthinking

3. Construct viable arguments andcritique the reasoning of others

Mathematics LessonMathematics Lesson Rigorous Task/Problem Classroom Discourse

In order for a problem/task to be rigorous Meaningful classroom discourse is it must meet the following criteria: imperative to extend student’s thinking•The problem/task has important, useful and connect mathematical ideas.mathematics embedded in it. i.e. Where is “Discourse includes ways of representing,it in the standard course of study? thinking, talking, agreeing, and disagreeing;•The problem/task requires higher-level the way ideas are exchanged and what the thinking and problem solving. ideas entail; and as being shaped by the tasks•The problem/task contributes to the in which students engage as well as by theconceptual development of students. nature of the learning environment.”•The problem/task creates an opportunity -NCTMfor the teacher to assess what his or herstudents are learning and where they areexperiencing difficulty.

THANK YOU……We looked forward to THANK YOU……We looked forward to seeing you again at training #2seeing you again at training #2