Nate FranzSuperv isor of Mathematics

November 19, 2013

Instructional Leadership in Mathematics:

Promoting Understanding

Austin’s Butterfly

What type of feedback does Austin receive on his butterfly?

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Feedback and Critique• The specific changes in his drawings can be linked to very specific feedback from peers, illuminating the need for critique to be targeted and specific.• An inspirational model of the power of perseverance and revision to improve quality

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Guiding Questions

On what specific features - the ones that really make a difference in how students come to view mathematics and what they ultimately learn - of a math classroom should we target

our feedback?

What are ways that instructional leaders can create an environment that expects and

supports this type of feedback?

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TIMSS Video Study

Goals Effort to go beyond cross-national achievement data to

focus on the underlying processes that produce achievement

Scope of Project Germany, Japan, Australia, Hong Kong, Czech Republic,

Netherlands 638 randomly selected, eighth grade lessons

throughout the entire year.

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TIMSS Video Study: Describe the Data

What do you see?Gather as much information as possible from

the dataIdentify where observations are being made –

e.g., “One page one in the second column…”Avoid judgments about quality or

interpretation

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Types of Math ProblemsTeacher Implementation of Making Connections Problems

Austral

ia

Czech R

epub

lic

Hong K

ong

Japan

Netherl

ands

United

State

s0

20406080

100

6177 84

4157

69

15 16 13

54

24 17

Using ProceduresMaking Connections

Austral

ia

Czech R

epub

lic

Hong K

ong

Japan

Netherl

ands

United

State

s0

102030405060708090

3116 18 20 19

59

8

52 46 4837

0

Using ProceduresMaking Connections

TIMSS Video Study7

TIMSS Video Study: Interpret the Data

What does the data suggest? What are the assumptions we can make about features of math classrooms?

Make sense of what the data says and why.Find as many interpretations as possible and

evaluate them against the kind and quality of evidence.

Think broadly and creatively. Assume that the data, no matter how confusing, makes sense to some people, your job is to see what they may see.

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Types of Math Problems

Austral

ia

Czech R

epub

lic

Hong K

ong

Japan

Netherl

ands

United

State

s0

20406080

100

6177 84

4157

69

15 16 13

54

24 17

Using ProceduresMaking Connections

Austral

ia

Czech R

epub

lic

Hong K

ong

Japan

Netherl

ands

United

State

s0

102030405060708090

3116 18 20 19

59

8

52 46 4837

0

Using ProceduresMaking Connections

TIMSS Video Study9

Teacher Implementation of Making Connections Problems

Dimensions and Core Features of Classrooms that Promote Understanding

Dimensions Core Features

Nature of Classroom Tasks

Make mathematics problematicConnect with where students areLeave behind something of mathematical value

Role of the TeacherSelect Tasks with goals in mindShare essential informationEstablish classroom culture

Social Culture of the Classroom

Ideas and methods are valuedStudents choose and share their methodsMistakes are learning sites for everyoneCorrectness resides in mathematical argument

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Making Sense: Teaching and Learning Mathematics with Understanding (Hiebert, 1997)

Math Department Focus Areas

High Level Tasks Task predicts performance; if we increase the cognitive

demand of the work students are engaged in, their learning will increase.

Fluency All students need to compute knowledge of

procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.

Conceptual Models and Tools Leverage vertical coherence; the connections in math

will strengthen understanding and allow for reason and intuition.

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Task Sorting Guide

Lower-level demands (memorization)Lower-level demands (procedures

without connections)High-level demands (procedures with

connections)High-level demands (doing

mathematics)

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Selecting and Creating Mathematics Tasks: From Research to Practice (Smith and Stein, 1998)

Knowing a “Good” Task When You See One

1. Working as a table, sort the cards into the four groups.

2. Develop criteria for each category.

3. Record your thoughts on Demand/Criteria slip.

4. Be prepared for discussion and reflection.

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Final Product

• Tasks sorted into the four levels of cognitive demand

• Criteria identified for each level of demand

• Note or Post-It with the tasks for each group.

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Knowing a Good Task When You See One

1. Does a particular feature indicate that the task has a certain level of cognitive demand?

2. Can you think of other factors that might make a task appear to be high level on the surface but that actually only require recall of memorized information?

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Knowing a Good Task When You See One

Features of tasks

Memorization

Procedures without

ConnectionsProcedures with

ConnectionsDoing

Mathematics

H F D A B E G C

Manipulatives * * * *Calculators *

Diagram * * * *Real-World

Context * *Symbolic/Abstract

Multi-steps, actions * *

Explanation * * * *

Textbook-like * * *

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Lower-Level Demands (Memorization)

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Mathematical Tasks as a Framework for Reflection (Smith and Stein, 1999)

Lower-Level Demands (Procedures without Connections)

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Mathematical Tasks as a Framework for Reflection (Smith and Stein, 1999)

High-Level Demands (Procedures with Connections)

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Mathematical Tasks as a Framework for Reflection (Smith and Stein, 1999)

High-Level Demands (Doing Mathematics)

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Mathematical Tasks as a Framework for Reflection (Smith and Stein, 1999)

Guiding Questions

On what specific features - the ones that really make a difference in how students come to view mathematics and what they ultimately learn - of a math classroom should we target

our feedback?

What are ways that instructional leaders can create an environment that expects and

supports this type of feedback?

21

Dimensions and Core Features of Classrooms that Promote Understanding

Dimensions Core Features

Nature of Classroom Tasks

Make mathematics problematicConnect with where students areLeave behind something of mathematical value

Role of the TeacherSelect Tasks with goals in mindShare essential informationEstablish classroom culture

Social Culture of the Classroom

Ideas and methods are valuedStudents choose and share their methodsMistakes are learning sites for everyoneCorrectness resides in mathematical argument

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Task Sorting Guide

Lower-level demands (memorization)Lower-level demands (procedures

without connections)High-level demands (procedures with

connections)High-level demands (doing

mathematics)

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Selecting and Creating Mathematics Tasks: From Research to Practice (Smith and Stein, 1998)

Flexible Approaches Promote Access24

Concrete PictorialAbstract

1/10th 1/100th 1/1,000th

1/10th 1/100th 1/1,000th

1

1

2

3

3

0.123 __ 0.13

Addition

Janet picked 3 daisies and 2 sunflowers from her garden. How many total flowers did Janet

pick from her garden?

Janet picked a total of __________ flowers.

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Janet’s

flowers

D DD S S ?

Subtraction

A total of 438 people were at the concert. There were 213 children and the rest were

adults. How many adults were at the concert?

There were _______ adults at the concert.

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People at

concert

213 438?C A

Multiplication

Ling put 3 photos on each page of her album. If there were 6 pages, how many photos did Ling

put in her album?

Ling put a total of _______ photos in her album.

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Ling’s photos 3 ?

pg.

3

pg.

3

pg.

3

pg.

3

pg.

3

pg.

Multiplication (Comparison)

Itty and Bitty each did jumping jacks. Itty did 8 jumping jacks. Bitty did 4 times as many

jumping jacks as Itty. How many jumping jacks did Itty and Bitty do altogether?

Itty and Bitty each did _______ jumping jacks altogether.

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Itty’s jumpin

g jacks

8?

Bitty’s jumpin

g jacks

8 8 8 8

Division

Simon arranged chairs in the gym for an assembly. He put 42 chairs into 6 equal rows.

How many chairs were in each row?

There were _______ chairs in each row.

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Chairs in

rows? 42

Fractions

Abu earned $30 mowing lawns on Saturday. He spent half of the money on a new CD, and he spent 1/3 of

the remaining money on lunch. Does he have enough money to also buy a bike attachment that costs

$12.98?

Abu has _______ left.

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Abu’s Money CD $30

$5 $5 $5

Ratio

The ratio of Ty’s books to Ling’s books is 3 : 4. Ty has 60 books. If Ty buys another 5 books, what will be the new ratio of Ty’s

books to Ling’s books?

The new ratio of Ty’s books to Ling’s books is _______.

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Ty’s Books

60

Ling’s Books

20 20 20

202020 20

Dimensions and Core Features of Classrooms that Promote Understanding

Dimensions Core Features

Nature of Classroom Tasks

Make mathematics problematicConnect with where students areLeave behind something of mathematical value

Role of the TeacherSelect Tasks with goals in mindShare essential informationEstablish classroom culture

Social Culture of the Classroom

Ideas and methods are valuedStudents choose and share their methodsMistakes are learning sites for everyoneCorrectness resides in mathematical argument

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First Draft Final Draft

Back to Austin33

Guiding Questions

On what specific features - the ones that really make a difference in how students come to view mathematics and what they ultimately learn - of a math classroom should we target

our feedback?

What are ways that instructional leaders can create an environment that expects and

supports this type of feedback?

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Homework and Instructional Levers

• Identify one instructional lever that you are interested in improving

• Collect artifacts (protocols, templates…) that you are using

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Content and

Curriculum

Assessment and DDI

Planning and Instruction

Observation and Feedback

Professional Development

One Leadership Path

Review curricular resourceDeep dive into the content – progressions,

standards, connectionsPlan tasks, make instructional decisions and

determine student outcomesExecute lessonReflect on student thinking and work

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K-5 Measurement Progressions

Overview

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K-5 Measurement Progressions

Grade 2

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K-5 Measurement Progressions

Grade 3

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Objective: Investigate and use the formulas for area and perimeter of rectangles

What features of the task provide students to make connections?

What features might you add to this task to give all students a chance to use reason and intuition to meet the objective?

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Grade 4 Module 3 Lesson 1 Problem Set

My Changes

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Actual Student Thinking

Student A

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Actual Student Thinking

Student B

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Actual Student Thinking

Student C

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Actual Student Thinking

Student D

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Coaching Touchstones

Was there opportunity for the students to learn? Why and why not?

What evidence was there that the mathematics was in fact learned?

What worked and was worthy of praise?What didn’t work and why?What opportunities were missed?What growth nugget can I end with or leave with

the teacher?

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Tilling the Soil for the CCSSM: Ten Essential Math Leader Mindsets (Steve Leinwand, 2012)

Become Aware of Cultural Routines We can only change teaching by using methods known

to change culture. Primary among these methods is the analysis of practice, which brings cultural routines to awareness so that teachers can consciously evaluate and improve them. A recent study by Hill and Ball (in press) of a large-scale professional development program found that analysis of classroom practice was one of three factors predicting growth of teachers' content knowledge.

Analysis of classroom practice plays several important roles. It gives teachers the opportunity to analyze how teaching affects learning and to examine closely those cases in which learning does not occur. It also gives teachers the skills they need to integrate new ideas into their own practice. For example, by analyzing videotaped examples of other teachers implementing making connections problems, teachers can identify the techniques used to implement such problems, as well as the way in which teachers embed these techniques within the flow of a lesson.

Attempts to implement reform without analysis of practice are not likely to succeed.

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Focus on the Details of Teaching, Not Teachers Most current efforts to improve the quality of

teaching focus on the teacher: how the profession can recruit more qualified teachers and how we can remedy deficiencies in the knowledge of current teachers. The focus on teachers has some merit, of course, but we believe that a focus on the improvement of teaching—the methods that teachers use in the classroom—will yield greater returns.

A focus on teaching must avoid the temptation to consider only the superficial aspects of teaching: the organization, tools, curriculum content, and textbooks. The cultural activity of teaching—the ways in which the teacher and students interact about the subject—can be more powerful than the curriculum materials that teachers use. As Figure 2 shows, even when the curriculum includes potentially rich problems, U.S. teachers use their traditional cultural teaching routines to transform the problems and reduce their instructional potential. We must find a way to change not just individual teachers, but the culture of teaching itself.

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Sherin and van Es (2005) have proposed three key components of teachers’ ability to notice classroom interactions:

identification of what is important in teaching situations

connection of specific classroom interactions with the broader concepts and principles of teaching (e.g., equity, classroom norms, mathematical topics)

use of knowledge about the teaching context (e.g., public school, grade eight mathematics) to reason about the situation.

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