“Common Core State Standards for Mathematics 9-12

Administrator’s Academy 1169”

Pippen ConsultingRandy and Sue Pippen2011-12pippenconsulting@aol.com

Warm-Up You have three playing cards lying face up,

side by side. A five is just to the right of a two, a five is just to the left of a two, a spade is just to the left of a club, and a spade is just to the right of a spade. What are two possibilities for the three cards?

Be ready to discuss your thinking!

Find a shoulder partner that is not in your school or district – move if you have to.

Introduce yourselves to each other:◦ Name, position, what you hope to learn today.

On a signal, tell the group what your partner told you.

Introductions

Turn to partner and discuss

1. Does it look different at elementary, middle and high school?

2. Is this design effective? What is our evidence that it is? What is our evidence that it is not?

3. How long have we used this model?

What does the traditional math lesson look like?

Rate Your Knowledge

Signal your familiarity with the new Illinois State Standards for Mathematics (Common Core State Standards) by showing a signal of 1 to 5 with 1 being the lowest.

I D E A M O C

- understand that the Common Core Math State Standards are the new Illinois State Math Standards and will be the basis for the Math State Assessments for grades 9-12;

-learn for evaluation purposes that the new Common Core Math State Standards involve content and practice standards - what mathematics is to be taught and assessed, and what instructional practices are expected to be used for grades 9-12;

-examine how grades 9-12 math instruction and assessment must change in order to teach and assess for understanding, making sense, and what to monitor through evaluation; and

-analyze the differences between the grades 9-12 scope and sequence of the old Illinois Learning Standards

Participants will be able to:

Workshop Goals• Relate the New Common Core State Standards

to the Illinois Standards and the upcoming change in State testing.

• Relate the new Mathematics Practice Standards to the way instruction should look with the CCSSM.

• Familiarize administrators with the instructional changes required for students to learn with depth, understanding and making sense of the mathematics.

• Relate the differences in the old Illinois Math Standards and the new Illinois Math Standards (CCSSM).

• Develop a plan to update staff on the key components of the Content and Practice Standards and how they will be assessed.

Major Ideas of the CCSSM Fewer, higher, more focused Benchmarked Internationally Equal emphasis of understanding and skills Much more specific than old Illinois Learning

Standards Emphasis on number early on, learning

trajectories develop through the grades Highly visual and connected with multiple

representations of functions: graphs/verbal/symbolic/numeric

Major Content Differences Emphasis on arithmetic and number patterns translating to

algebra Congruence and similarity based on transformations Resurgence of constructions, but in a variety of ways Algebra 1, Geometry, and Algebra 2 for all students Modeling, modeling, modeling or “What’s it good for?” Precalculus only for students who will take calculus Not all students should take calculus – STEM standards (+) A variety of fourth year courses No longer push for more students in the 8th grade taking

high school algebra

Currently sending too many underprepared students to algebra at the 8th grade

Program may not be equivalent to high school due to time constraints of middle school, may not have a secondary-math- certified teacher

There cannot be any skipping in CCSSM There are other ways to accelerate (p. 81

Appendix A) Not all students need calculus, therefore do

not need to accelerate at all.

8th Grade Algebra

Understand the connections between proportional relationships, lines, and linear equations.

5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Eighth Grade Expressions and Equations

Analyze and solve linear equations and pairs of simultaneous linear equations.

7. Solve linear equations in one variable.

a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

Eighth Grade Expressions and Equations

No ISAT or PSAE after 2013-2014.

May be pilot items in ISAT in 2012-2014.

Some areas tested by current state tests will no longer be tested in new design.

NCLB has not been reauthorized nor made any adjustments for CCSS. Many states are refusing to continue with NCLB.

A waiver is to be available to states who meet the criteria.to be released in September

New Assessment Design

FocusedASSESSMENT4

• Speaking• Listening

25%

FocusedASSESSMENT

1• ELA• Math

50%

FocusedASSESSMENT

2• ELA• Math

90%

END OF YEARCOMPREHENSIVE ASSESSMENT

75%

FocusedASSESSMENT

3• ELA• Math

PARTNERSHIP RESOURCE CENTER: Digital library of released items, formative assessments, model curriculum frameworks, curriculum resources, student and educator tutorials and practice tests, scoring training modules, and professional development materials

Summative assessment for accountability

Required, but not used tor accountability

The PARCC System – Initial Design

English Language Arts and Mathematics, Grades 3 - 11

Partnership for Assessment of Readiness for College and Careers (PARCC)

Governing Board States Participating States

1. Create high-quality assessments 2. Build a pathway to college and career

readiness for all students3. Support educators in the classroom4. Develop 21st century, technology-based

assessments5. Advance accountability at all levels

The PARCC Goals

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Priority Purposes of PARCC Assessments:

1. Determine whether students are college- and career-ready or on track

2. Assess the full range of the Common Core Standards, including standards that are difficult to measure

3. Measure the full range of student performance, including the performance high and low performing students

4. Provide data during the academic year to inform instruction, interventions and professional development

5. Provide data for accountability, including measures of growth

6. Incorporate innovative approaches throughout the system

Summative Assessment Components:◦ Performance-Based Assessment (PBA) administered as close to the

end of the school year as possible. The ELA/literacy PBA will focus on writing effectively when analyzing text. The mathematics PBA will focus on applying skills, concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools.

◦ End-of-Year Assessment (EOY) administered after approx. 90% of the school year. The ELA/literacy EOY will focus on reading comprehension The math EOY will be comprised of innovative, machine-scorable items

Formative Assessment Components:◦ Early Assessment designed to be an indicator of student knowledge

and skills so that instruction, supports and professional development can be tailored to meet student needs

◦ Mid-Year Assessment comprised of performance-based items and tasks, with an emphasis on hard-to-measure standards. After study, individual states may consider including as a summative component

Goal #1: Create High Quality Assessments

The PARCC assessments will allow us to make important claims about students’ knowledge and

skills. In English Language Arts/Literacy, whether students:

◦ Can Read and Comprehend Complex Literary and Informational Text◦ Can Write Effectively When Analyzing Text◦ Have attained overall proficiency in ELA/literacy

In Mathematics, whether students:◦ Have mastered knowledge and skills in highlighted domains (e.g.

domain of highest importance for a particular grade level – number/ fractions in grade 4; proportional reasoning and ratios in grade 6)

◦ Have attained overall proficiency in mathematics

Goal #1: Create High Quality Assessments

Goal #1: Create High-Quality Assessments – New Design

End-of-Year Assessment

• Innovative, computer-based items

Performance-BasedAssessment (PBA)• Extended tasks• Applications of

concepts and skills

Summative assessment for accountability

Formative assessment

Early Assessment• Early indicator of

student knowledge and skills to inform instruction, supports, and PD

ELA/Literacy

• Speaking• Listening

Flexible

Mid-Year Assessment

• Performance-based

• Emphasis on hard to measure standards

• Potentially summative

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Goal #2: Build a Pathway to College and Career Readiness for All Students

K-2 3-8 High School

K-2 formative assessment being

developed, aligned

to the PARCC system

Timely student achievement data showing students,

parents and educators whether

ALL students are on-track to college and

career readiness

ONGOING STUDENT SUPPORTS/INTERVENTIONS

College readiness score to identify who is

ready for college-

level coursework

SUCCESS IN FIRST-YEAR,

CREDIT-BEARING,

POSTSECONDARY

COURSEWORK

Targeted interventions & supports:

• 12th-grade bridge courses

• PD for educators

23

Goal #3: Support Educators in the Classroom

PROFESSIONAL DEVELOPMENT MODULES

INSTRUCTIONAL TOOLS TO SUPPORT

IMPLEMENTATION

EDUCATOR-LED TRAINING TO SUPPORT “PEER-TO-PEER”

TRAININGTIMELY STUDENT

ACHIEVEMENT DATA

K-12 Educator

24

Goal #4: Develop 21st Century, Technology-Based AssessmentsPARCC’s assessment will be computer-based and leverage technology in a range of ways to: Item Development

◦ Develop innovative tasks that engage students in the assessment process

Administration◦ Reduce paperwork, increase security, reduce shipping/receiving &

storage◦ Increase access to and provision of accommodations for SWDs and

ELLs Scoring

◦ Make scoring more efficient by combining human and automated approaches

Reporting◦ Produce timely reports of students performance throughout the

year to inform instructional, interventions, and professional development

25

Goal #4: Develop 21st Century, Technology-Based Assessments

PARCC assessments will be purposefully designed to generate valid, reliable and timely data, including measures of growth, for various accountability uses including:◦ School and district effectiveness◦ Educator effectiveness◦ Student placement into college-credit bearing courses◦ Comparisons with other state and international

benchmarks PARCC assessments will be designed for other

accountability uses as states deem appropriate

PARCC Timeline

26

Sept. 2011

Development phase begins

Sept. 2012

First year field testing and

related research and

data collection begins

Sept. 2013

Second year field testing begins and

related research and

data collection continues

Sept. 2014

Full administration

of PARCC assessments

begins

Oct. 2010

Launch and design phase

begins

Summer 2015

Set achievement

levels, including

college-ready performance

levels

27

Key Challenges for PARCC

Implementation Challenges

Estimating costs over time, including long-term budgetary planning

Transitioning to the new assessments at the classroom level

Ensuring long-term sustainability

Policy Challenges Student supports

and interventions Accountability High school

course requirements

College admissions/ placement

Perceptions about what these assessments can do

Technical Challenges• Developing an

interoperable technology platform

• Transitioning to a computer-based assessment system

• Developing and implementing automated scoring systems and processes

• Identifying effective, innovative item types

Reason for New Assessment Design Change

Cost effectiveness in a difficult economy The three summative through-course

assessments could dictate the scope and sequence of the curriculum limiting local flexibility (not federal government right)

The potential that the required three through-course assessments would disrupt the instructional program on, and in preparation for, testing days

Intended to ensure results will be reported in categories consistent with the CCSS.

Separate scores in ELA for reading and writing as well as an overall score indicating on track to college and career readiness.

Separate score in a “highlighted domain” that reflects the CCSS’s emphasis at each grade level (e.g., fractions in grade 4, rations and proportional relationships at grade 6), as well as an overall math score indicating on track to college readiness.

Measures student growth over a full academic year or course Provides data during the academic year to inform instruction,

interventions and professional development activities. Accessible to all students including disabled and ELL Must be approved by the US Department of Education

Highlighted Domains - PARCC Grade or

HS CategoryHighlighted Domains

K CC1 OA2 NBT3 OA4 NF5 NF6 RP. EE7 RP, NS8 EE, G

HS-NQ RNHS-A SSE, REIHS-F IF, BFHS-M No separate scoreHS-G CO, GPEHS-SP ID

Brain Break! Listen to directions See what it looks like Stand up and try it

Comparison of Content StandardsOld Illinois Learning

StandardsNCTM Standards Common Core State

StandardsNumber Number Sense Number and Quantity

ModelingMeasurement Measurement

Algebra Algebra Algebra

Functions

Modeling

Geometry Geometry GeometryModeling

Probability and Statistics

Probability and Statistics

Probability and StatisticsModeling

Comparison of Process StandardsOld Illinois Learning

StandardsNCTM Standards Common Core State Standards

Solve Problems Problem Solving Model with Mathematics

Make sense of problems and persevere in solving them

Look for and express regularity in repeated reasoning

Look for and make use of structure

Working on Teams

Using Technology Use appropriate tools strategically

Communicating Communication Construct viable arguments and critique the reasoning of others

Look for and express regularity in repeated reasoning

Attend to precision (language)

Making Connections Connections

Representation Attend to precision

Reasoning and Proof Reason abstractly and quantitatively

Construct viable arguments and critique the reasoning of others

Progressions Kindergarten 1 2 3 4 5 6 7 8 HS

Counting and Cardinality

Number and

QuantityNumber and Operations in Base Ten Ratios and Proportionality

Number and Operations - Fractions The Number System

Operations and Algebraic Thinking

Expressions and Equations Algebra

Functions Functions

Geometry Geometry Geometry

Measurement and Data Statistics and ProbabilityStatistics

and Probability

EARLY ELEMENTARY

LATE ELEMENTARY

MIDDLE/JUNIOR HIGH SCHOOL

EARLY HIGH SCHOOL

LATE HIGH SCHOOL

6.A.1a Identify whole numbers and compare them using the symbols <, >, or = and the words “less than”, “greater than”, or “equal to”, applying counting, grouping and place value concepts.

6.A.2 Compare and order whole numbers, fractions and decimals using concrete materi als, drawings and mathematical symbols.

6.A.3 Represent fractions, decimals, per centages, exponents and scientific notation in equivalent forms.

6.A.4 Identify and apply the associative, commutative, distributive and identity proper ties of real numbers, including special numbers such as pi and square roots.

6.A.5 Perform addition, subtraction and multiplication of complex numbers and graph the results in the complex plane.

6.A.1b Identify and model fractions using concrete materials and pictorial representations.

Number Goal – ILS (Old)

N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.

N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Reason quantitatively and use units to solve problems.

1a. How do you solve 3x + 1 = -14 ?

1b. Why did you do it the way you did?

Switch roles

2a. How do you graph y = ½ x -3?

2b. Why did you do it the way you did?

Explain to your partner…

EARLY ELEMENTARY

LATE ELEMENTARY

MIDDLE/JUNIOR HIGH SCHOOL

EARLY HIGH SCHOOL

LATE HIGH SCHOOL

8.D.1 Find the unknown numbers in whole-number addition, subtraction, multiplication and division situations.

8.D.2 Solve linear equations involving whole numbers.

8.D.3a Solve problems using numeric, graphic or symbolic representations of varia bles, expressions, equations and inequalities.

8.D.4 Formulate and solve linear and quadratic equations and linear inequalities algebraically and investigate nonlinear inequalities using graphs, tables, calculators and computers.

8.D.5 Formulate and solve nonlinear equations and systems including problems involving inverse variation and exponential and logarithmic growth and decay.

8.D.3b Propose and solve problems using proportions, formulas and linear functions.

8.D.3c Apply properties of powers, perfect squares and square roots.

Algebra Goal - OLD ILS

A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Understand solving equations as aprocess of reasoning and explain the reasoning. (CCSSM Algebra)

9.B.1a Identify and describe characteristics, similarities and differences of geometric shapes.

9.B.2 Compare geometric figures and determine their properties including parallel, perpendicular, similar, congruent and line symmetry.

9.B.3 Identify, describe, classify and compare two- and three- dimensional geometric figures and models according to their properties.

9.B.4 Recognize and apply relationships within and among geometric figures.

9.B.5 Construct and use two- and three-dimensional models of objects that have practical applications (e.g., blueprints, topo graphical maps, scale models).

9.B.1b Sort, classify and compare familiar shapes.

9.B.1c Identify lines of symmetry in simple figures and construct symmetrical figures using various concrete materials.

Geometry Goal – Old ILSEARLY ELEMENTARY

LATE ELEMENTARY

MIDDLE/JUNIOR HIGH SCHOOL

EARLY HIGH SCHOOL

LATE HIGH SCHOOL

Geometry - CCSSM G.CO.1 Know precise definitions of angle, circle, perpendicular line,

parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Why is Change Needed? There is a train. It leaves a station an hour later than a plane

flying overhead, flying in the opposite direction. The number of the train is a 3-digit number whose tens digit

is 3 more than its units digit. The conductor of the train is half as old as the train was

when the conductor was a third as old, just a third as old.  The conductor’s niece and nephew are on the train. They

head toward the club car at the back of the train to buy mixed nuts; some of the nuts are $1.79 a pound and some are $2.25 a pound.

 They have quarters, dimes and nickels in their pockets to pay for the nuts.

 The niece starts first and walks at 2 miles per hour and the nephew starts later and walks at 3 miles per hour.

 How long will it take them to get to the back of the train if they walk together?

Enuf said?

No Numbers Warm-up

• If you know the width of a lawn mower in inches, how can you find how many square yards of lawn it cuts in running a certain number of feet?

▫Problems Without Figures▫Gillan, 1909

Traditional Path or Integrated Path

Same fifteen units – distributed by course

Illinois will have to choose one or the other to determine testing

Challenges: Materials for either path

Texts: May say they are aligned, probably not

High School – Appendix A

Common Core State StandardsAlgebra I Unit 1 – Relationships

Between Quantities and Reasoning with Equations

Unit 2 – Linear and Exponential Relationships

Unit 3 – Descriptive Statistics

Unit 4 - Expressions and Equations

Unit 5 – Quadratic Functions and Modeling

Mathematics I Unit 1 – Relationships

Between Quantities Unit 2 – Linear and

Exponential Relationships Unit 3 – Reasoning with

Equations Unit 4 – Descriptive

Statistics Unit 5 – Congruence, Proof

and Constructions Unit 6 – Connecting Algebra

and Geometry through Coordinates

Common Core State StandardsGeometry Unit 1 - Congruence, Proof,

and Constructions Unit 2 - Similarity, Proof

and Trigonometry Unit 3 - Extending to Three

Dimensions Unit 4 - Connecting

Algebra and Geometry through Coordinates

Unit 5 - Circles with and Without Coordinates

Unit 6 - Applications of Probability

Mathematics II Unit 1 – Extending the

Number System Unit 2 - Quadratic Functions

and Modeling Unit 3 – Expressions and

Equations Unit 4 – Applications of

Probability Unit 5 – Similarity, Right

Triangle Trigonometry and Proof

Unit 6 – Circles With and Without Coordinates

Common Core State StandardsAlgebra II

Unit 1 – Polynomial, Rational and Radical Relationships

Unit 2 – Trigonometric Functions

Unit 3 – Modeling with Functions

Unit 4 – Inferences and Conclusions from Data

Mathematics III

Unit 1 – Inferences and Conclusions from Data

Unit 2 – Polynomial, Rational and Radical Relationships

Unit 3 – Trigonometry of (+)General Triangles and Trigonometric Functions

Unit 4 – Mathematical Modeling

More algebra at the eighth grade means a different algebra in high school, more technology for both

Geometry must be built upon grade school transformations – most books are not written that way

More Probability and Stats in all high school courses

Advanced Algebra has less content but more depth than previous courses, more technology

Major Changes at the High School

Turn to your shoulder partner and talk about what you see regarding the new and old ILS – specifically, talk about implications for instruction

Signal to start, signal to stop (about 2 minutes).

Whole Group Sharing

Discussion – Partner Talk

Brain Break! Listen to directions See what it looks like Stand up and try it

Create the Vision of Quality Math Instruction

What is Mathematics Proficiency?

Two sources: Strands of Proficiency from Adding It Up and Mathematical Practice Standards (CCSSM)

52

Underlying Frameworks

Strands of Mathematical Proficiency

Strategic Competence

Adaptive Reasoning

Conceptual Understanding

Productive Disposition

Procedural Fluency

NRC (2001). Adding It Up. Washington, D.C.: National Academies Press.

53

• Conceptual Understanding – comprehension of mathematical concepts, operations, and relations

• Procedural Fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately

• Strategic Competence – ability to formulate, represent, and solve mathematical problems

• Adaptive Reasoning – capacity for logical thought, reflection, explanation, and justification

• Productive Disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

Strands of Mathematical Proficiency

Standards for Mathematical Practice

1. In pairs, review the Standards for Mathematical Practice. Take the standards two at a time, one for each of you, then share what you read. Return to whole group to discuss. Then back to pairs, repeat.

2. When finished with all eight, discuss a new insight you had into the practices.

Mathematics Practice Standards

◦ Make sense of problems and persevere in solving them

◦ Reason abstractly and quantitatively◦ Construct viable arguments and critique the reasoning

of others◦ Model with Mathematics◦ Use appropriate tools strategically◦ Attend to precision◦ Look for and make use of structure◦ Look for and express regularity in repeated reasoning

NCTM Vision

Are we there yet? What will it take?

What Should a Math Classroom Look Like? Brainstorming

Handout – What Should I look for in a Math Classroom?

Best practice instructionLESS MORE Lecturing Students passive Value on student

silence Worksheet/seatwork “Coverage” Competition Rote memorization Tracking/pullouts Reliance on outside

tests

Experiential/hands-on Active Learning Student conversations Higher order thinking Deeper study of fewer

topics Choice for students Student responsibility Help within classroom Heterogeneous

grouping Teacher’s evaluation of

learning

On Motivation It is not something

you do to others Maximum

motivation occurs when the person believes he has autonomy, mastery and purpose

Control leads to compliance, autonomy leads to engagement

Mastery is the desire to get better and better at something that matters

Choice plays into autonomy – turn homework into “homelearning”

“Now-that” rewards instead of “if-then” rewards, non-tangible are best

Seven Reasons Carrots and Sticks Don’t Often Work

1. They can extinguish intrinsic motivation

2. They can diminish performance.

3. They can crush creativity

4. They can crowd out good behavior

5. They can encourage cheating, shortcuts and unethical behavior

6. They can become addictive

7. They can foster short-term thinking

From Drive, Daniel Pink

Praise effort and strategy, not intelligence

Make praise specific, not generalPraise in private, one-on-oneOffer praise only when there is a good reason for it

Praise

What is the Role of Curriculum?

“A curriculum is more than a collection of activities; it must be coherent, focused on important mathematics, and well articulated across the grades.” NCTM Principles and Standards for School Mathematics 2000

The curriculum is not the textbook!

NCTM Focal Points – a good elementary resource

Common Written Curriculum – Clear Objectives Common Core State Standards

What is the Role of Assessment? “Assessment should support the learning of

important mathematics and furnish useful information to both teachers and students.” NCTM Principles and Standards, 2000.

Aligned to Objectives and Could be Arranged by Objectives

Common Major Assessments Frequent Informal Assessments with

Immediate Feedback Feedback for Guiding Instruction and Goal

Setting

Sample

64

Administrative Issues Effective Professional development:

◦ Develops teachers’ knowledge of math content, students and how they learn mathematics, effective instructional and assessment practices

◦ Models examples of high-quality mathematics teaching and learning

◦ Allows teachers to reflect on their practice and student learning in their classroom

◦ Allows teachers to collaborate and share experience with colleagues

◦ Connects to a comprehensive long-term plan that includes student achievement

Discussion

Effective Practices

Video

Discussion: What is the teacher doing, what are the students doing?

Handout – During the Observation

Discussion

Observing and Evaluating

Activity to Demonstrate New Perspective on Learning Seating people at tables

◦ If each table can seat 8 people with three on a side and one at each end.

◦ When tables are pushed together end to end, people can sit on each side and only at each end.

◦ How many people can be seated at 2 tables end to end? 3 tables, end to end 5 tables, end to end n tables, end to end

Focus on Meaning

Emphasis on the mathematical meaning

Having students constructing their meaning

Making connections between mathematics and other subject matter areas

Building on student meanings and student understandings

Learning New Concepts and Skills While Solving Problems Having students solve problems without

prior or concurrent skill development. Allowing students to explore and develop

their own algorithms Having students learn skill development

through problem solving, conjecturing and verifying.

Drill on isolated skills can hinder making sense of them later.

“The joy of the task is its own reward.”

Teach Mathematics Right the First Time – Steve Leinwand Students taught procedures tend to resist

new ideas and appeared to apply procedures without understanding. (Kieran, 1984)

“Initial rote learning of a concept can create interference to later meaningful learning” (Pesek and Kirshner, 2000)

Based on an article in Educational Leadership,

Video Clip

Video

Who is doing the work?

What is the engagement level of the students?

Concrete Materials

Hands-on experiences enable students to construct their own meanings.

Teachers must be knowledgeable in the use of concrete materials.

Using the same material to teach different ideas help shorten the time it takes to see connections between mathematical ideas.

Do not limit to demonstrations. Students must see the two-way

relationship between the concrete materials and the notation used to represent it.

Try This One: 2x – 4 = 8

2x - 4 = 8

Add 4 to each side and remove zero pairs.Arrange the tiles into two equal groups on both

sides of the mat.Answer?

Student Use of Calculators Changes the content, methods, and skill

requirements Enables more high-level questions. Actively involves students through asking

questions, conjecturing and exploring – lots of exploring with discussion about what is happening and why

Positive effects on graphing ability, conceptual understanding of graphs, and relating graphs to other representations.

Students using graphing calculators are more flexible with strategies, have greater perseverance, and trying to understand concepts.

Teach through tasks instead of “telling”

Employ a variety of student thinking

Recognize and value different methods

May include manipulatives, but most of all relies on thinking and recording thinking

The Role of Tasks

Nature of Classroom Tasks

Make mathematics problematic – you have not already taught them how

Connect with where students are – varied levels of entry

Leave behind something of mathematical value – mathematical learning

Role of the Teacher Select tasks with goals in mind Share essential information Establish classroom culture

◦ Ideas and methods are valued◦ Students choose and share their methods◦ Mistakes are learning sites for everyone ◦ Correctness resides in the mathematical

argument

What Are Mathematical Tasks?

Mathematical tasks are a set of problems or a single complex problem the purpose of which is to focus students’ attention on a particular mathematical idea.

Why Focus on Mathematical Tasks?

Tasks form the basis for students’ opportunities to learn what mathematics is and how one does it;

Tasks influence learners by directing their attention to particular aspects of content and by specifying ways to process information;

The level and kind of thinking required by mathematical instructional tasks influences what students learn; and

Differences in the level and kind of thinking of tasks used by different teachers, schools, and districts, is a major source of inequity in students’ opportunities to learn mathematics.

Golden Crown Task - Sample

The King asks Archimedes if his crown is made from pure gold.

He knows that the crown is either pure gold or it may have some silver in it.

Archimedes figures out that the volume of the crown is 125 cm3 and that its mass is 1.8 kilograms.

He also knows that 1 kilogram of gold has a volume of about 50 cm3 and 1 kilogram of silver has a volume of about 100 cm3.

1. Is the crown pure gold? Explain how you know. 2. If the crown is not pure gold, then how much

silver is in it? Show all your work.

A professional development resource

Released in April from NCTM

Aligns well with the CCSSM Mathematical Practices

Five Practices Book

Tasks in the Classroom Choose the task Work it out and anticipate student methods Conduct a classroom discussion to clarify the task,

but not direct the students to a solution or method, close reading

Monitor the work and identify which groups are using which methods or new methods

Select and record which groups will present Sequence the presentations for maximum

discussion Connect the ideas with a whole-class discussion

From the 5 Practices book

Adapt classroom problems – choose from the end of the unit before teaching the unit – make it an application problem.

Consult the Internet – see sources at the end of the PowerPoint

Focus on the math you want them to learn

Where to Find Tasks

Talk Formats

To get better participation in classroom conversations, move between three formats:◦ Whole-class discussion – before a task, after a

task◦ Small-group discussion – time limit, specific

directions on what they are to do/discuss/produce◦ Partner talk – short time limit to get more thinking

when the whole-class discussion stalls out, specific directions on what they are to discuss (30 seconds)

Talk to a partner

Orchestrating Classroom Talk Five productive talk moves

◦ Revoicing (teacher)◦ Repeating (student)◦ Reasoning - Agree/disagree and why (student)◦ Adding on (student)◦ Wait time (teacher)

Examples Revoicing: “So you’re saying it’s an odd number?” Repeating: “Can you repeat what he just said in

your own words?” Reasoning: “Miranda, do you agree or disagree

with what Paul just said?” Adding on: “Would someone like to add

something more to this?” Wait time: Wait beyond the time for a few

students to raise their hands. Wait for the reluctant participants to think and offer an explanation. (10 seconds or more)

Implementing Classroom Talk Five steps to implementing classroom talk

◦ Set the classroom climate, respectful and supportive

◦ Focus the talk on the mathematics◦ Provide for equitable participation◦ Explain your expectations for the new forms of

talk and why talk in math is important◦ Try only one challenging new thing at a time

Video of Talk Moves Identify talk moves in the video as the teacher

launches a lesson on linear equations.

http://www.insidemathematics.org/index.php/classroom-video-visits/public-lessons-comparing-linear-functions/269-comparing-linear-functions-problem-2-part-a?phpMyAdmin=NqJS1x3gaJqDM-1-8LXtX3WJ4e8

Discussion

Second Video - from book: 6.2

Who’s Doing the Work? Is the teacher always the one talking? Do students present solutions? Do students work together? Do students converse about mathematics

with each other or with the teacher? Are students building their own meaning or

is the teacher dispensing it?

Changing Perspectives on Learning and Teaching

All learning, except for simple rote memorization, requires the learner to actively construct meaning

Students’ prior understanding of and thoughts about a topic or concept before instruction exert a tremendous influence on what they learn during instruction

The teacher’s primary goal is to generate a change in the learner’s cognitive structure or way of viewing and organizing the world

Because learning is a process of active construction by the learner, the teacher cannot do the work of learning

Learning in cooperation with others is an important source of motivation, support, modeling and coaching

Number off by 7s Go to the numbered poster with a marker. Write implications for instructional leaders

according to the topic at the top of the poster.

At signal, move to next poster and repeat.

Summary Discussion and Reflection

Wrap-It-Up Carousel

Required of the academy

Your plan should be how to disseminate the information you learned about today.

It must be submitted to Donna at the St. Clair ROE to be entered into the system for you.

Write a Plan

Workshop Goals• Relate the New Common Core State Standards

to the Illinois Standards and the upcoming change in State testing.

• Relate the new Mathematics Practice Standards to the way instruction should look with the CCSSM.

• Familiarize administrators with the instructional changes required for students to learn with depth, understanding and making sense of the mathematics.

• Relate the differences in the old Illinois Math Standards and the new Illinois Math Standards (CCSSM).

• Develop a plan to update staff on the key components of the Content and Practice Standards and how they will be assessed.

What was most valuable to you today?

Contact info: pippenconsulting@aol.com

If you want a copy of this PowerPoint: http://dl.dropbox.com/u/26625625/2011%201169%20AA.ppt

Final Thoughts

Task Resources www.nctm.org Illuminations www.insidemathematics.org www.nctm.org Navigations Books and Focus

Books Coming: illustrativemathematics.org Coming: www.mathedleadership.org – Great

Tasks and More (NCSM website) www.mathedleadership.org - Common Core State

Standards (CCSS) Mathematics Curriculum Materials Analysis Project

Challenge problems in texts Enrichment activities – maybe Word problems not taught yet

Other Resources Five Practices for Orchestrating Productive Mathematics

Discussions, Smith and Stein, NCTM, 2011. Classroom Discussions, Using Math Talk to Help

Students Learn, Chapin, Math Solutions, 2009. Handbook of Research on Improving Student

Achievement, Third Edition, Gordon Cawelti, Editor, Educational Research Service, 2004.

Common Core Standards, NGA, CCSSO, 2010 Annenberg Media Videos Drive, The Surprising Truth about What Motivates Us,

Daniel Pink, 2009.

Conferences NCSM Annual Meeting, Philadelphia, April 2012 NCTM Annual Meeting, Philadelphia, April 2012