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Unless footers note otherwise, all pages are copyrighted to © Toncheff 2013 and are REPRODUCIBLE. • solution-tree.com

How Do You Collaboratively Plan for CCSS-M (6–12)?

Mona Toncheff toncheff@phoenixunion.org http://puhsdmath.blogspot.com

WE

Paradigm Shifts• Professional development

– Ongoing collaborative team learning• Instruction

– Teaching for conceptual understanding as well as procedural fluency

• Content– Focus, coherence, rigor; conceptual

understanding and procedural fluency• Assessment

– Multifaceted process; emphasis on formative assessment

• Intervention– Required, not invitational

2

Today’s Learning Targets • I can examine criteria and effective

lesson planning and design on a unit-by-unit basis.

• I can create great tasks that develop student access to CCSS-M.

• I can define the learning progressions of the K–12 CCSS-M.

Three Big Ideas

1. Focus on student learning

2. Focus on collaboration

3. Focus on results

--DuFour, DuFour, Eaker, & Many, Learning by Doing (2010)

3

Four PLC Questions1. What do we expect students to learn?2. How will we know students learned it?

3. What will we do when students do not learn?

4. What will we do when students do learn?

--DuFour, DuFour, Eaker, & Many, Learning by Doing (2010)

Seven Stages of Teacher CollaborationStage 1: Filling the timeWhat exactly are we supposed to do?

Stage 2: Sharing personal practiceWhat is everyone doing in their classroom?

Stage 3: Planning, planning, planningWhat should we be teaching and how do we lighten the load for each other?

4

Stage 4: Developing common assessmentsHow do you know students learned?What does mastery look like?

Stage 5: Analyzing student learningAre students learning what they are supposed to be learning?

Seven Stages of Teacher Collaboration

Stage 6: Adapting instruction to student needsHow can we adjust instruction to help struggling students and those who exceed expectations?

Stage 7: Reflecting on instructionWhich Mathematical Practices are most effective with our students for this lesson or unit?

Seven Stages of Teacher Collaboration

5

Where Are We Now?

Think about your current course based or grade level collaborative teams.

Which Stage? Scan and decide.

How Do We Get to Stage 7?

• Norms for collaboration • Shared vision of mathematics

Action orientation: Teams do and produce stuff on a unit-by-unitbasis.

6

Brief Excerpt from Common Core Mathematics in a PLC at Work™ Dr. Timothy D. Kanold tkanold.blogspot.com

Seeking Stage Seven as a Team Graham and Ferriter (2008) offer a useful framework that details seven stages of collaborative team development. Adapted for our purposes, the stage at which teams fall is directly correlated to each team’s level of effective collaboration. Table 1 highlights these seven stages. Table 1: The Seven Stages of Teacher Collaboration Diagnostic Tool

Stage Questions That Define This Stage

Stage 1: Filling the time What exactly are we supposed to do? Why are we meeting? Is this going to be worth my time?

Stage 2: Sharing personal practice What is everyone doing in his or her classroom? What are some of your favorite problems you use for this unit?

Stage 3: Planning, planning, planning What content should we be teaching, and how should we pace this unit? How do we lighten the load for each other?

Stage 4: Developing common assessments

How do you know students learned? What does mastery look like? What does student proficiency look like?

Stage 5: Analyzing student learning Are students learning what they are supposed to be learning? What does it mean for students to demonstrate understanding of the learning targets?

Stage 6: Adapting instruction to student needs

How can we adjust instruction to help those students struggling and those exceeding expectations?

Stage 7: Reflecting on instruction Which of our instructional and assessment practices are most effective with our students?

Visit go.solution-tree.com/commoncore for a reproducible version of this table.

11Excerpted from Common Core Mathematics in a PLC at Work™ series

© Solution Tree Press 2012. Do not duplicate.1787

Common Core State Standards for MathematicsTwo type of standards:• Standards for Mathematical

Practice

• Standards for Mathematical Content

How Familiar Are You With Standards for Mathematical Practice?

Rate your knowledgeon a scale of

5 (high) to 1 (low)

8

Students Standards for Mathematical Practice

“The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.

These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education.”

--Common Core State Standards Initiative (2010)

9

How Do We Develop Student’s Mathematical Practice?“Shift to include within daily lesson plans intentional strategies to teach mathematics in different ways – in ways that focus on the process of learning and developing deep student understanding of the content.”

—Kanold (Ed.), Common Core Mathematics in a PLC at Work™, K– 2 (2012)

Collaborative Team Work

Develop a common understanding of the Standards for Mathematical Practice:1. What is the intent of this

Mathematical Practice?

2. What teacher actions facilitate this Mathematical Practice?

3. What evidence is there that students are demonstrating this Mathematical Practice?

10

How Do We Develop Student’s Mathematical Practice?Intentional strategies:• Purposeful planning

• Rich mathematical tasks that develop each practice

What is your experience with collaborative lesson design?

Strengths Weaknesses 

How Do We Develop Student’s Mathematical Practice?

11

What Questions Do You and Your Team Ask Each Other When Planning?

Collaborative Lesson Design Activity

7.RP Ratios and Proportional RelationshipsAnalyze proportional relationships and use them to solve real-world and mathematical problems.

F-LE: Construct and compare linear, quadratic, and exponential models and solve problems.

Grade-7 Lesson Grade-9 Lesson

12

R E PRO DUCI B LE

Common Core Mathematics in a PLC at WorkTM, Leader’s Guide © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/commoncore to download this page.

Table 2.1: Elements of an Effective Mathematics Classroom Lesson Design

Probing Questions for Effective Lesson Design Reflection

1. Lesson Context: Learning Targets

Procedural Fluency and Conceptual Understanding Balancing

What is the learning target for the lesson? How does it connect to the bigger focus of the unit?

What evidence will be used to determine the level of student learning of the target?

Are conceptual understanding and procedural fluency appropriately balanced?

How will you formatively assess student conceptual understanding of the mathematics concepts and of the procedural skill?

What meaningful application or model can you use?

Which CCSS Mathematical Practices will be emphasized during this lesson?

2. Lesson Process: High-Cognitive-Demand Tasks

Planning Student Discourse and Engagement

What tasks will be used that create an a-ha student moment and leave “mathematical residue” (insights into the mathematical structure of concepts) regardless of content type at a high-cognitive-demand level?

How will you ensure the task is accessible to all students while still maintaining a high cognitive demand for students?

What strategic mathematical tools will be used during the lesson?

page 1 of 2

R E PRO DUCI B LE

Common Core Mathematics in a PLC at WorkTM, Leader’s Guide © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/commoncore to download this page.

Probing Questions for Effective Lesson Design Reflection

2. Lesson Process: High-Cognitive-Demand Tasks

(continued)

How will each lesson example be presented and sequenced to build mathematical reasoning connected to prior student knowledge?

What are the assessing and advancing questions you might ask during guided, independent, or group practice? What are anticipated student responses to the examples or tasks?

How might technology and student attention to precision play a role in the student lesson experience?

3. Introduction, Daily Review, and Closure

What activity will be used to immediately engage students at the beginning of the class period?

How can the daily review be used to provide brief, meaningful feedback on homework? (Five minutes maximum)

How will the students summarize the lesson learning targets and the key vocabulary words?

4. Homework How does the homework assignment provide variety and meaning to the students—including long-term review and questions—that balance procedural fluency with conceptual understanding?

page 2 of 2

Collaborative Lesson Design Activity • CCSS‐M Standards  

Grade‐7 Ratios and Proportional Relationships 

Analyze proportional relationships and use them to solve real‐world and mathematical problems.  

7.RP.A.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour. 

7.RP.A.2 Recognize and represent proportional relationships between quantities. 

7.RP.A.2a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. 

7.RP.A.2b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 

7.RP.A.2c Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. 

7.RP.A.2d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1,r) where r is the unit rate. 

7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. 

 

Grade‐9 Linear, Quadratic, and Exponential Models 

Construct and compare linear, quadratic, and exponential models and solve problems. 

HSF‐LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. 

HSF‐LE.A.1a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. 

HSF‐LE.A.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 

HSF‐LE.A.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. 

HSF‐LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input‐output pairs (include reading these from a table). 

HSF‐LE.A.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. 

HSF‐LE.A.4 For exponential models, express as a logarithm the solution toabct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. 

15(Source: Common Core State Standards Initiative, 2010) © National Governors Association Center for Best Practicesand Council of Chief State School Officers 2010. All rights reserved. Do not duplicate.

CCSS Mathematical Practices LP Tool

—Kanold & Larson, Common Core Mathematics in a PLC at Work™, Leader’s Guide (2012)

Collaboration Begins With the End in Mind …

• Grade 7: RP cluster• Grade 9: F-LE cluster with exponential

functions • Review your standards and complete

the table.

16

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Common Core Mathematics in a PLC at Work™, Grades 3–5, page 97© 2012 Solution Tree Press · solution-tree.com

Do not duplicate. 17

R E PRO DUCI B LE

Common Core Mathematics in a PLC at WorkTM, Grades K–2 © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/commoncore to download this page.

Unit: Date: Lesson:

Learning target: As a result of today’s class, students will be able to

Formative assessment: How will students be expected to demonstrate mastery of the learning target during in-class checks for understanding?

Probing Questions for Differentiation on Mathematical Tasks

Assessing Questions

(Create questions to scaffold instruction for students who are “stuck” during the lesson or the lesson tasks.)

Advancing Questions

(Create questions to further learning for students who are ready to advance beyond the learning target.)

Targeted Standard for Mathematical Practice:

(Describe the intent of this Mathematical Practice and how it relates to the learning target.)

Tasks

(The number of tasks may vary from lesson to lesson.) What Will the Teacher Be Doing?

What Will the Students Be Doing?

(How will students be actively engaged in each part of the lesson? )

Beginning-of-Class Routines

How does the warm-up activity connect to students’ prior knowledge?

Figure 2.11: CCSS Mathematical Practices Lesson-Planning Tool

page 1 of 2

18

R E PRO DUCI B LE

Common Core Mathematics in a PLC at WorkTM, Grades K–2 © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/commoncore to download this page.

Tasks

(The number of tasks may vary from lesson to lesson.) What Will the Teacher Be Doing?

What Will the Students Be Doing?

(How will students be actively engaged in each part of the lesson? )

Task 1

How will the learning target be introduced?

Task 2

How will the task develop student sense making and reasoning?

Task 3

How will the task require student conjectures and communication?

Closure

How will student questions and reflections be elicited in the summary of the lesson? How will students’ understanding of the learning target be determined?

page 2 of 2

19

Photo Sizes Math Task, Grade 7  

  

Photographs come in several standard print sizes.  Most common print sizes are 4x6, 5x7,  and 8x10.  (Note: The dimensions are given in inches.)    Does a proportional relationship exist between these print sizes? Justify your answer.         Follow‐Up Extension Questions: 1. 3.5x5 used to be a popular photo size.  How does it relate to the other standard sizes? 

     

2. Some consider wallet size photographs 2x3, while others say they are 2.5x3.5.   Based on your findings, what are the dimensions of a standard size wallet photo? 

    

 3. Explain what will happen if you take a 4x6 photo and enlarge it to an 8x10.       

 7.RP.A.2 Recognize and represent proportional relationships between quantities. 2a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin 

20

Algebra 1-2 Name: _________________________________________

Unit 3 Day 3-4

Directions:

Guided Practice:

1.

2.

x f(x)

-2 0.12

0 3

2 75

4 1875

6 46875

My little sister, Savannah,

is 3 years old and has a

piggy bank she wants to

fill. She started with five

pennies and every day

when I come home from

school, I give her my three

pennies that are left over

from my lunch money to

add to her piggy bank.

A. Identify the pattern of change in each of the relations.

Equal differences over equal intervals

Equal factors over equal intervals

Neither

B. Be prepared to describe how you found the pattern of change.

C. Identify the type of function in each of the relations.

Linear Function

Exponential Function

Neither

Type of pattern of change __________________________________________

How I found the pattern of change:

Type of function ___________________________________________________

Type of pattern of change __________________________________________

How I found the pattern of change:

Type of function ___________________________________________________

21

2 3y x

3xy

3.

4.

5.

6.

Type of pattern of change __________________________________________

How I found the pattern of change:

Type of function ___________________________________________________

Type of pattern of change __________________________________________

How I found the pattern of change:

Type of function ___________________________________________________

Type of pattern of change __________________________________________

How I found the pattern of change:

Type of function ___________________________________________________

Type of pattern of change __________________________________________

How I found the pattern of change:

Type of function ___________________________________________________

[

22

7.

8.

9.

The pattern of change

in the perimeter of the

figures from one step

to the next.

The pattern of change

in the area of the

figures from one step

to the next.

Type of pattern of change __________________________________________

How I found the pattern of change:

Type of function ___________________________________________________

Type of pattern of change __________________________________________

How I found the pattern of change:

Type of function ___________________________________________________

Type of pattern of change __________________________________________

How I found the pattern of change:

Type of function ___________________________________________________

23

10.

11.

12.

The algae population

in a pond doubles

every year.

The change in the

height of a ball from

one bounce to the

next bounce is 4/5 of

its previous height.

The ball was first

dropped from a height

of 80 feet.

Type of pattern of change __________________________________________

How I found the pattern of change:

Type of function ___________________________________________________

Type of pattern of change __________________________________________

How I found the pattern of change:

Type of function ___________________________________________________

Type of pattern of change __________________________________________

How I found the pattern of change:

Type of function ___________________________________________________

24

• What are learning targets for the lesson?

• How would you assess the targets?

Collaboration Begins With the End in Mind …

What Learning Experiences Will Develop These Targets?

25

How Will We Create Access to the Tasks?

What MPs Will Be Targeted?

26

Handouts for Teachers Improving Learning Through Questioning

Handout 3: Five principles for effective questioning

1. Plan to use questions that encourage thinking and reasoning

Really  effective  questions  are  planned  beforehand.  It  is  helpful  to  plan  sequences  of  questions  that  build  on  and  extend  students’  thinking.  A  good  questioner,  of  course,  remains  flexible  and  allows  time  to  follow  up  responses.      

Beginning an inquiry

• What  do  you  already  know  that  might  be  useful  here?  • What  sort  of  diagram  might  be  helpful?  • Can  you  invent  a  simple  notation  for  this?  • How  can  you  simplify  this  problem?  • What  is  known  and  what  is  unknown?  • What  assumptions  might  we  make?  

Progressing with an inquiry

• Where  have  you  seen  something  like  this  before?  • What  is  fixed  here,  and  what  can  we  change?    • What  is  the  same  and  what  is  different  here?  • What  would  happen  if  I  changed  this  ...  to  this  ...  ?  • Is  this  approach  going  anywhere?  • What  will  you  do  when  you  get  that  answer?  • This  is  just  a  special  case  of  ...  what?  • Can  you  form  any  hypotheses?  • Can  you  think  of  any  counterexamples?  • What  mistakes  have  we  made?  • Can  you  suggest  a  different  way  of  doing  this?  • What  conclusions  can  you  make  from  this  data?  • How  can  we  check  this  calculation  without  doing  it  all  again?  • What  is  a  sensible  way  to  record  this?  

Interpreting and evaluating the results of an inquiry

• How  can  you  best  display  your  data?    • Is  it  better  to  use  this  type  of  chart  or  that  one?  Why?  • What  patterns  can  you  see  in  this  data?  • What  reasons  might  there  be  for  these  patterns?  • Can  you  give  me  a  convincing  argument  for  that  statement?  • Do  you  think  that  answer  is  reasonable?  Why?  • How  can  you  be  100%  sure  that  is  true?  Convince  me!  • What  do  you  think  of  Anne’s  argument?  Why?  • Which  method  might  be  best  to  use  here?  Why?  

Communicating conclusions and reflecting

• What  method  did  you  use?  • What  other  methods  have  you  considered?  • Which  of  your  methods  was  the  best?  Why?  • Which  method  was  the  quickest?  • Where  have  you  seen  a  problem  like  this  before?    • What  methods  did  you  use  last  time?  Would  they  have  worked  here?  • What  helpful  strategies  have  you  learned  for  next  time?  

 

http://map.mathshell.org/static/draft/pd/modules/4_Questioning/html/index.htm27

Mathematical Practices “Look Fors”

Goals of Assessment“We must ensure that tests measure what is of value, not just what is easy to test.

“If we want students to investigate, explore, and discover, assessment must not measure just mimicry mathematics.”

—National Research Council, Everybody Counts (2000)

28

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mea

ning

of s

ymbo

ls, c

aref

ully

spec

ify u

nits

of m

easu

re, a

nd p

rovi

de

accu

rate

labe

ls.

C

alcu

late

acc

urat

ely

and

effic

ient

ly, e

xpre

ssin

g nu

mer

ical

ans

wer

s with

a d

egre

e of

pre

cisi

on.

P

rovi

de c

aref

ully

form

ulat

ed e

xpla

natio

ns.

L

abel

acc

urat

ely

whe

n m

easu

ring

and

grap

hing

. C

omm

ents

:

E

mph

asiz

e th

e im

porta

nce

of p

reci

se c

omm

unic

atio

n by

enc

oura

ging

stud

ents

to

focu

s on

clar

ity o

f the

def

initi

ons,

nota

tion,

and

voc

abul

ary

to c

onve

y th

eir

reas

onin

g.

E

ncou

rage

acc

urac

y an

d ef

ficie

ncy

in c

ompu

tatio

n an

d pr

oble

m-b

ased

so

lutio

ns, e

xpre

ssin

g nu

mer

ical

ans

wer

s, da

ta, a

nd/o

r mea

sure

men

ts w

ith a

de

gree

of p

reci

sion

app

ropr

iate

for t

he c

onte

xt o

f the

pro

blem

. C

omm

ents

:  

Reasoning and explaining

2.

Rea

son

abst

ract

ly a

nd

quan

titat

ivel

y.

M

ake

sens

e of

qua

ntiti

es a

nd re

latio

nshi

ps in

pro

blem

situ

atio

ns.

R

epre

sent

abs

tract

situ

atio

ns sy

mbo

lical

ly a

nd u

nder

stan

d th

e m

eani

ng

of q

uant

ities

.

Cre

ate

a co

here

nt re

pres

enta

tion

of th

e pr

oble

m a

t han

d.

C

onsi

der t

he u

nits

invo

lved

.

Fle

xibl

y us

e pr

oper

ties o

f ope

ratio

ns.

Com

men

ts:

F

acili

tate

opp

ortu

nitie

s for

stud

ents

to d

iscu

ss o

r use

repr

esen

tatio

ns to

mak

e se

nse

of q

uant

ities

and

thei

r rel

atio

nshi

ps.

E

ncou

rage

the

flexi

ble

use

of p

rope

rties

of o

pera

tions

, obj

ects

, and

solu

tion

stra

tegi

es w

hen

solv

ing

prob

lem

s.

Pro

vide

opp

ortu

nitie

s for

stud

ents

to d

econ

text

ualiz

e (a

bstra

ct a

situ

atio

n)

and/

or c

onte

xtua

lize

(iden

tify

refe

rent

s for

sym

bols

invo

lved

) the

mat

hem

atic

s th

ey a

re le

arni

ng.

Com

men

ts:

3.

Con

stru

ct

viab

le

argu

men

ts a

nd

criti

que

the

reas

onin

g of

ot

hers

.

U

se d

efin

ition

s and

pre

viou

sly

esta

blis

hed

caus

es a

nd e

ffec

ts (r

esul

ts) i

n co

nstru

ctin

g ar

gum

ents

.

Mak

e co

njec

ture

s and

use

cou

nter

exam

ples

to b

uild

a lo

gica

l pro

gres

sion

of

stat

emen

ts to

exp

lore

and

supp

ort i

deas

.

Com

mun

icat

e an

d de

fend

mat

hem

atic

al re

ason

ing

usin

g ob

ject

s, dr

awin

gs,

diag

ram

s, an

d/or

act

ions

.

Lis

ten

to o

r rea

d th

e ar

gum

ents

of o

ther

s.

Dec

ide

if th

e ar

gum

ents

of o

ther

s mak

e se

nse

and

ask

prob

ing

ques

tions

to

clar

ify o

r im

prov

e th

e ar

gum

ents

. C

omm

ents

:

P

rovi

de a

nd o

rche

stra

te o

ppor

tuni

ties f

or st

uden

ts to

list

en to

the

solu

tion

stra

tegi

es o

f oth

ers,

disc

uss a

ltern

ativ

e so

lutio

ns, a

nd d

efen

d th

eir i

deas

.

Ask

hig

her-

orde

r que

stio

ns th

at e

ncou

rage

stud

ents

to d

efen

d th

eir i

deas

.

Pro

vide

pro

mpt

s tha

t enc

oura

ge st

uden

ts to

thin

k cr

itica

lly a

bout

the

mat

hem

atic

s the

y ar

e le

arni

ng.

Com

men

ts: 

© Fennell 2012. solution-tree.comReproducible.

REPRODUCIBLE

29

Mat

hem

atic

s Pra

ctic

es

Stud

ents

T

each

ers

Modeling and using tools 4.

Mod

el w

ith

mat

hem

atic

s.

A

pply

prio

r kno

wle

dge

to so

lve

real

-wor

ld p

robl

ems.

I

dent

ify im

porta

nt q

uant

ities

and

map

thei

r rel

atio

nshi

ps u

sing

such

tool

s as

diag

ram

s, tw

o-w

ay ta

bles

, gra

phs,

flow

cha

rts, a

nd/o

r for

mul

as.

U

se a

ssum

ptio

ns a

nd a

ppro

xim

atio

ns to

mak

e a

prob

lem

sim

pler

.

Che

ck to

see

if an

ans

wer

mak

es se

nse

with

in th

e co

ntex

t of a

situ

atio

n an

d ch

ange

a m

odel

whe

n ne

cess

ary.

C

omm

ents

:

U

se m

athe

mat

ical

mod

els a

ppro

pria

te fo

r the

focu

s of t

he le

sson

.

Enc

oura

ge st

uden

t use

of d

evel

opm

enta

lly a

nd c

onte

nt-a

ppro

pria

te

mat

hem

atic

al m

odel

s (e.

g., v

aria

bles

, equ

atio

ns, c

oord

inat

e gr

ids)

.

Rem

ind

stud

ents

that

a m

athe

mat

ical

mod

el u

sed

to re

pres

ent a

pro

blem

’s

solu

tion

is a

wor

k in

pro

gres

s, an

d m

ay b

e re

vise

d as

nee

ded.

C

omm

ents

:

5. U

se

appr

opri

ate

tool

s st

rate

gica

lly.

M

ake

soun

d de

cisi

ons a

bout

the

use

of sp

ecifi

c to

ols (

exam

ples

mig

ht in

clud

e ca

lcul

ator

, con

cret

e m

odel

s, di

gita

l tec

hnol

ogie

s, pe

ncil/

pape

r, ru

ler,

com

pass

, an

d pr

otra

ctor

).

Use

tech

nolo

gica

l too

ls to

vis

ualiz

e th

e re

sults

of a

ssum

ptio

ns, e

xplo

re

cons

eque

nces

, and

com

pare

pre

dica

tions

with

dat

a.

I

dent

ify re

leva

nt e

xter

nal m

ath

reso

urce

s (di

gita

l con

tent

on

a w

ebsi

te) a

nd u

se

them

to p

ose

or so

lve

prob

lem

s.

Use

tech

nolo

gica

l too

ls to

exp

lore

and

dee

pen

unde

rsta

ndin

g of

con

cept

s. C

omm

ents

:

U

se a

ppro

pria

te p

hysi

cal a

nd/o

r dig

ital t

ools

to re

pres

ent,

expl

ore,

and

dee

pen

stud

ent u

nder

stan

ding

.

Hel

p st

uden

ts m

ake

soun

d de

cisi

ons c

once

rnin

g th

e us

e of

spec

ific

tool

s ap

prop

riate

for t

he g

rade

-leve

l and

con

tent

focu

s of t

he le

sson

.

Pro

vide

acc

ess t

o m

ater

ials

, mod

els,

tool

s, an

d/or

tech

nolo

gy‐ b

ased

reso

urce

s th

at a

ssis

t stu

dent

s in

mak

ing

conj

ectu

res n

eces

sary

for s

olvi

ng p

robl

ems.

Com

men

ts:

Seeing structure and generalizing

7. L

ook

for

and

mak

e us

e of

st

ruct

ure.

L

ook

for p

atte

rns o

r stru

ctur

e, re

cogn

izin

g th

at q

uant

ities

can

be

repr

esen

ted

in

diff

eren

t way

s.

Rec

ogni

ze th

e si

gnifi

canc

e in

con

cept

s and

mod

els a

nd u

se th

e pa

ttern

s or

stru

ctur

e fo

r sol

ving

rela

ted

prob

lem

s.

Vie

w c

ompl

icat

ed q

uant

ities

bot

h as

sing

le o

bjec

ts o

r com

posi

tions

of s

ever

al

obje

cts a

nd u

se o

pera

tions

to m

ake

sens

e of

pro

blem

s. C

omm

ents

:

E

ngag

e st

uden

ts in

dis

cuss

ions

em

phas

izin

g re

latio

nshi

ps b

etw

een

parti

cula

r to

pics

with

in a

con

tent

dom

ain

or a

cros

s con

tent

dom

ains

.

Rec

ogni

ze th

at th

e qu

antit

ativ

e re

latio

nshi

ps m

odel

ed b

y op

erat

ions

and

thei

r pr

oper

ties r

emai

n im

porta

nt re

gard

less

of t

he o

pera

tiona

l foc

us o

f a le

sson

.

Pro

vide

act

iviti

es in

whi

ch st

uden

ts d

emon

stra

te th

eir f

lexi

bilit

y in

re

pres

entin

g m

athe

mat

ics i

n a

num

ber o

f way

s, e.

g., 7

6 =

(7 x

10)

+ 6

; di

scus

sing

type

s of q

uadr

ilate

rals

, and

so o

n.

Com

men

ts:

8.

Loo

k fo

r an

d ex

pres

s re

gula

rity

in

repe

ated

re

ason

ing.

N

otic

e re

peat

ed c

alcu

latio

ns a

nd lo

ok fo

r gen

eral

met

hods

and

shor

tcut

s.

Con

tinua

lly e

valu

ate

the

reas

onab

lene

ss o

f int

erm

edia

te re

sults

(com

parin

g es

timat

es),

whi

le a

ttend

ing

to d

etai

ls, a

nd m

ake

gene

raliz

atio

ns b

ased

on

findi

ngs.

Com

men

ts:

E

ngag

e st

uden

ts in

dis

cuss

ion

rela

ted

to re

peat

ed re

ason

ing

that

may

occ

ur in

a

prob

lem

’s so

lutio

n.

D

raw

atte

ntio

n to

the

prer

equi

site

step

s nec

essa

ry to

con

side

r whe

n so

lvin

g a

prob

lem

.

Urg

e st

uden

ts to

con

tinua

lly e

valu

ate

the

reas

onab

lene

ss o

f the

ir re

sults

. C

omm

ents

:

© Fennell 2012. solution-tree.comReproducible.

REPRODUCIBLE

30

Rich Learning Experiences

What decides the cognitive demand of a task?

It is decided not by whether it is a hard problem, but rather by the complexity of reasoning required by the student.

(Kanold, Briars, & Fennel, What Principals Need to Know About Teaching and Learning Mathematics (2011)

31

Hess’ Cognitive Rigor Matrix & Curricular Examples: Applying Webb’s Depth-of-Knowledge Levels to Bloom’s Cognitive Process Dimensions – Math/Science

© 2009 Karin Hess permission to reproduce is given when authorship is fully cited khess@nciea.org

Revised Bloom’sTaxonomy

Webb’s DOK Level 1Recall & Reproduction

Webb’s DOK Level 2Skills & Concepts

Webb’s DOK Level 3Strategic Thinking/ Reasoning

Webb’s DOK Level 4Extended Thinking

RememberRetrieve knowledge fromlong-term memory,recognize, recall, locate,identify

o Recall, observe, & recognizefacts, principles, properties

o Recall/ identify conversionsamong representations ornumbers (e.g., customary andmetric measures)

UnderstandConstruct meaning, clarify,paraphrase, represent,translate, illustrate, giveexamples, classify,categorize, summarize,generalize, infer a logicalconclusion (such as fromexamples given), predict,compare/contrast, match likeideas, explain, constructmodels

o Evaluate an expressiono Locate points on a grid or

number on number lineo Solve a one-step problemo Represent math relationships in

words, pictures, or symbolso Read, write, compare decimals

in scientific notation

o Specify and explain relationships(e.g., non-examples/examples;cause-effect)

o Make and record observationso Explain steps followedo Summarize results or conceptso Make basic inferences or logical

predictions from data/observationso Use models /diagrams to represent

or explain mathematical conceptso Make and explain estimates

o Use concepts to solve non-routineproblems

o Explain, generalize, or connect ideasusing supporting evidence

o Make and justify conjectureso Explain thinking when more than

one response is possibleo Explain phenomena in terms of

concepts

o Relate mathematical orscientific concepts to othercontent areas, other domains,or other concepts

o Develop generalizations of theresults obtained and thestrategies used (frominvestigation or readings) andapply them to new problemsituations

ApplyCarry out or use a procedurein a given situation; carry out(apply to a familiar task), oruse (apply) to an unfamiliartask

o Follow simple procedures(recipe-type directions)

o Calculate, measure, apply a rule(e.g., rounding)

o Apply algorithm or formula (e.g.,area, perimeter)

o Solve linear equationso Make conversions among

representations or numbers, orwithin and between customaryand metric measures

o Select a procedure according tocriteria and perform it

o Solve routine problem applyingmultiple concepts or decision points

o Retrieve information from a table,graph, or figure and use it solve aproblem requiring multiple steps

o Translate between tables, graphs,words, and symbolic notations (e.g.,graph data from a table)

o Construct models given criteria

o Design investigation for a specificpurpose or research question

o Conduct a designed investigationo Use concepts to solve non-routine

problemso Use & show reasoning, planning,

and evidenceo Translate between problem &

symbolic notation when not a directtranslation

o Select or devise approachamong many alternatives tosolve a problem

o Conduct a project that specifiesa problem, identifies solutionpaths, solves the problem, andreports results

AnalyzeBreak into constituent parts,determine how parts relate,differentiate betweenrelevant-irrelevant,distinguish, focus, select,organize, outline, findcoherence, deconstruct

o Retrieve information from a tableor graph to answer a question

o Identify whether specificinformation is contained ingraphic representations (e.g.,table, graph, T-chart, diagram)

o Identify a pattern/trend

o Categorize, classify materials, data,figures based on characteristics

o Organize or order datao Compare/ contrast figures or datao Select appropriate graph and

organize & display datao Interpret data from a simple grapho Extend a pattern

o Compare information within oracross data sets or texts

o Analyze and draw conclusions fromdata, citing evidence

o Generalize a patterno Interpret data from complex grapho Analyze similarities/differences

between procedures or solutions

o Analyze multiple sources ofevidence

o analyze complex/abstractthemes

o Gather, analyze, and evaluateinformation

EvaluateMake judgments based oncriteria, check, detectinconsistencies or fallacies,judge, critique

o Cite evidence and develop a logicalargument for concepts or solutions

o Describe, compare, and contrastsolution methods

o Verify reasonableness of results

o Gather, analyze, & evaluateinformation to draw conclusions

o Apply understanding in a novelway, provide argument orjustification for the application

CreateReorganize elements intonew patterns/structures,generate, hypothesize,design, plan, construct,produce

o Brainstorm ideas, concepts, orperspectives related to a topic

o Generate conjectures or hypothesesbased on observations or priorknowledge and experience

o Synthesize information within onedata set, source, or text

o Formulate an original problem givena situation

o Develop a scientific/mathematicalmodel for a complex situation

o Synthesize information acrossmultiple sources or texts

o Design a mathematical modelto inform and solve a practicalor abstract situation

32

3232

Task Sort In a group of 34, sort the questions by DOK Level (1, 2, or 3).

There are three of each.

DOK ? 

1

2

3

4

Instructional Alignment What percent of your instructional materials develop each level of Depth of Knowledge?

DOK ? 

1 ?

2 ?

3 ?

4 ?

33

What Are Great Mathematical Tasks?

• Center on an interesting problem, offering several methods of solution

• Are directed at essential mathematical content as specified in the standards

• Require examination and perseverance (challenging to students)

• Beg for discussion, offering rich discourse on mathematics involved

• Build student understanding, following a clear set of learning expectations

• Warrant a summary look back with reflection and extension opportunities(www.mathedleadership.org/ccss/greattasks.html)

Five-Finger Rule

34

Sustained Implementation of CCSS requires four pursuits:

1. A thorough review of your current local assessments on a unit-by-unit basis

2. High-quality common assessments and the accurate scoring of those assessments

3. A robust formative assessment process for students and adults, using each assessment Instrument

4. Instruction that provides evidence of student understanding via the mathematical practices

K–8 Content Standards Connected to High School

35

Review Key Features of the Standards for Mathematical Content

Focus questions:Considering the new CCSS-M features that you have been discussing, what implications do these features have for your curriculum, instruction and assessment?

For your collaborative team work?

Where Do I Find Resources to Create a Coherent Learning Progression?

• PARCC model content frameworks

• Sample scope and sequences from the CCSS toolbox

• Learning progressions

• Evidence tables

36

Today’s Learning Targets • I can examine criteria and effective

lesson planning and design on a unit-by-unit basis.

• I can create great tasks that develops student access to CCSS-M.

• I can define the learning progressions of the K–12 CCSS-M.

End-of-Day Note Card Reflection

1. Has our work today caused you to consider or reconsider any aspects of your own thinking and/or practice? Explain.

2. Has our work today caused you to reconsider any aspects of your students’ mathematical learning? Explain.

3. What would you like more information about?

37

Questions?

Mona Toncheff toncheff@phoenixunion.org 

http://puhsdmath.blogspot.com  

WE

38