From Answer Finding to Doing

K–5 Mathematics Breakout SessionIntroduction to Professional Learning for New Teachers

August 24–26, 2015

Office of Curriculum, Instruction & Professional Learning

FACILITATOR NAMES

Welcome and Good Morning Please find your table.

Take a moment to introduce yourselves to the other new teachers in your group:• Your name• The borough in which you will be working• One topic in math that you are looking forward to exploring with

students this year

2

Agenda

Introductions & Norms

Hexagon Task Lesson: Tables & Chairs

Hexagon Task Lesson: Planning Highlights

Engaging, Exploring & Sharing: Shifts in Mathematics

Case Study: Teacher Actions that Support Doing Mathematics

Closing & Reflection

3

Session Outcomes Engage in a model lesson centering on a mathematics task

that is intellectually engaging and enjoyable and analyze pedagogical techniques that support student learning.

Experience and analyze a mathematics lesson structure that promotes sense making and students as doers of mathematics.

Become familiar with the eight Mathematics Teaching Practices from NCTM and use them to analyze a case study of a classroom engaging with a mathematics task.

Gain insight into a lesson structure and pedagogical practices that promote deeper mathematical learning and discuss the impact on lesson planning.

4

Norms for Collaboration

1. Promoting a Spirit of Inquiry and Balancing Advocacy

2. Pausing

3. Paraphrasing

4. Probing

5. Putting Ideas on the Table

6. Paying Attention to Self and Others

7. Presuming Positive Intentions

Adapted from CCE and the work of R. Garmston

5

Norms for Collaboration Read the description for each norm.

Identify a norm that:• supports you as a learner• you sometimes struggle with and would like to work on

Turn and talk:• Which norms did you choose and why?• How can norms be useful in the classroom?

6

7

Hexagon Task LessonTables & Chairs

Setting up for a banquet

How many chairs could I set up around this table?

How do you know?

Could you fit more? Fewer? How so?

Setting up a single table

How about at two tables?

But I want my guests to talk to each other!How many chairs could I set up with two tables placed next to each other like this?

I am planning a big party!I want to know how many chairs could be set up if I keep adding tables next to each other, like this:

And so on

13

Explore As you work on your own, consider:

• What mathematical ideas and strategies are you using?• How might a student approach this problem?

Work with your table groups to create a poster• Make sure your poster clearly illustrates your group’s thinking.• Please be prepared to discuss your group’s poster and thinking

with the entire group.

14

Reflect & Table Talk

Take a few moments to reflect:1. What stands out to you most about this classroom

experience?2. What teaching moves do you feel were effective in

promoting student learning and understanding?3. What features of the task itself helped to promote

discussion?

Table Talk

Group Share

15

Hexagon Task LessonPlanning Highlights

A few lesson planning considerations for:

16

Planning learning goals

Selecting a good task

Anticipating what students might say and do

Providing access to all students

Providing opportunities for students to think like mathematicians every day

Learning goals

17

Running out of time is a common challenge in the first year.

For each lesson, identify:

Need to know

Nice to know

How can learning goals be developed?

What are the standards?

What do they need for tomorrow’s lesson?

18

Has multiple entry points and various solution pathways

“Low floor, high ceiling”

Requires exploration of a mathematical relationship

Students explore the task first and then formalize and connect solution methods

Task that promotes “doing” mathematics

For more info, see the characteristics of mathematical tasks at four “Levels of Cognitive Demand” from Smith and Stein, 1998.

Anticipating student responses

19

Anticipated likely solution paths and student misconceptions:

Prepared questions to:

Support students with misconceptions

Advance student thinking

Produce “student work” in advance containing errors and solution paths

Providing access for all students

20

How does this lesson provide multiple entry points for different learners?

In what ways can the classroom environment support various learners during the lesson?

How about the teacher?

Grouping?

Thinking like mathematicians

21

As defined by the Standards for Mathematical Practice

Table talk: Which Standards for Mathematical Practice were evident in the lesson or do you see opportunities for?

Planning Tip: Look for ways to design your lessons to regularly engage students in the Standards for Mathematical Practice.

22

Engaging, Exploring & SharingShifts in Math Instruction

CCLS-M One Key Shift

Answer Finders Doers of Mathematics

• Most of the mathematical thinking happens before finding the answer.

• Success is identifying the proper procedure and applying it to solve a problem.

• Tasks are tools to assess student understanding.

• Tasks are tools to help students discover underlying mathematics in situations.

• Multiple approaches are essential for this discovery.

• Key mathematical work exists in connecting students’ ideas to one another and generalizing discoveries.

• The solution is one key step, but not the main purpose.

23

from Answer Finders to Doers of Mathematics

CCLS-M One Key Shiftfrom Answer Finders to Doers of Mathematics

Doing Mathematics

24

Finding the Answer

The Butterfly Effect

25

“Mechanical execution of procedures without understanding their mathematical basis often leads to bizarre results”

- Gary W Martin (2009)

What other mathematical “butterflies” have you experienced?

Traditional Lesson Structure for Mathematics

26

Homework Review• Students put answers on the board• Teachers show solution paths on request

Lecture• Concept explained in abstract• Teacher shows several examples• Students ask questions

Practice• Students assigned 1-20 odd for HW• 17 and 19 are tricky word problems• Answers in the back of the book

Assess• Chapter Test• Final Exam

Key Features• Teacher and text

book disseminate knowledge

• Application happens after basic skills

Metacognitive Modeling(I Do, We do, You do)

27

Do Now / Routines• Activates prior knowledge• May be topical or review

Mini-Lesson• Teacher models new content• Teacher thinking made explicit (think aloud)

Whole Class• Students try it with support from teacher and

one another• Whole class discussion• All voices heard

Independent Practice• Either in class or for HW

Key Features• Teacher thinking

is central and explicit

• Gradual Release• Assessment

happens throughout

Lesson Structure for Doing Mathematics(You Do, Ya’ll Do, We Do)

28

Engage – Whole Class• Develop enthusiasm• Access prior knowledge• Clarify conditions

Explore – Small Group• May begin independently• Develop solution path • Articulate thinking

Share – Whole Class• Share and connect solution paths• Identify and analyze errors• Concretize learning

Apply, Connect, Extend – Independent

Key Features• Sense-making

and assessment happen throughout

• Relatable, but not necessarily real world

• Focusing not funneling

Teacher and Student Roles

Traditional Meta-Cognitive Doers

Role of Teacher

Role of Student

29

Convey Content Make Teacher Thinking Explicit

Make Student Thinking Explicit

Connect Student Thinking

Solve Problem Sets

Learn to Think Like Teacher

Participate in a Community of Mathematicians

Connecting to the Hexagon Task Lesson

30

How did the Hexagon Task lesson reflect the “Doing Mathematics” lesson structure?

At your tables: What are students and the teacher doing during each phase? Note-taking tool to capture thoughts

Group Share

31

Case StudyTeacher Actions that Support Doing

Mathematics

Case Study

32

Exploring Linear Functions: The Case of Ms. Peterson• Note that the hexagon task used earlier was modified from

the version in this text.

As you read, consider: What does the teacher do to support student learning? How does this lesson support students doing mathematics?

Take a few minutes to share thoughts at your tables.

Group Share

Teaching Practices that support students in doing mathematics

33

“Mathematics Teaching Practices” from NCTM’s Principles to Actions

Read the Overview page

Turn and talk:Which Mathematics Teaching

Practices stand out to you most and why?

Which practices are you most curious about?

Teaching Practices and the Case Study

34

1. Read the “Teacher and Student Actions” table for your table’s specific Mathematics Teaching Practice

2. Re-read the Case Study through the lens of that single Mathematics Teaching Practice Capture any evidence of your Teaching Practice in the Case

Study as you re-read; please document line numbers.

3. In your table groups, discuss and chart the evidence that you discovered.

4. Group Share

35

Closing & Reflection

Reflection – Compass Points

“We don’t learn from experience, we learn from reflecting on experience” – John Dewey

36

N

S

EW

Something I am excited about doing in my classroom this year…

Something that I will share with my colleagues…

Something I am still wondering about…

Something new and/or noteworthy that I don’t want to forget…

Additional Resources

See the “Supplemental Resources” document for websites that have math tasks, classroom videos, lesson plans, professional learning materials, and other resources for implementing the Common Core in your classroom.

37

38

Thank You!