Moving Ahead with the Common Core Learning Standards

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Moving Ahead with the Common Core Learning Standards for Mathematics Professional Development | February

Moving Ahead with the Common Core Learning Standards for Mathematics Professional Development | February 17, 2012 RONALD SCHWARZ Math Specialist, America’s Choice, | Pearson School Achievement Services Slide 0 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. CFN 602

Block Stack Math Olympiad for Elementary and Middle Schools Slide 1 Copyright © 2010

Block Stack Math Olympiad for Elementary and Middle Schools Slide 1 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 25 layers of blocks are stacked; the top four layers are shown. Each layer has two fewer blocks than the layer below it. How many blocks are in all 25 layers?

 AGENDA Slide 2 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All

AGENDA Slide 2 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Standards for Math Content: Conceptual Shifts • What’s Different • Math Performance Tasks • Formative assessment • Resources 2

What are Standards? • Standards define what students should understand be able to do.

What are Standards? • Standards define what students should understand be able to do. • Any country’s standards are subject to periodic revision. • But math is more than a list of topics. 3 Slide 3 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • The US has been a jumble of 50 different state standards. Race to the bottom or the top?

DESPITE GAINS, ONLY 39% OF NYC 4 TH GRADERS AND 26% OF 8 TH

DESPITE GAINS, ONLY 39% OF NYC 4 TH GRADERS AND 26% OF 8 TH GRADERS ARE PROFICIENT ON NATIONAL MATH TESTS NAEP & NY STATE TEST RESULTS NYC MATH PERFORMANCE PERCENT AT OR ABOVE PROFICIENT 4 th Grade 2009 NAEP 2003 2009 2003 NY State Test 2009 NAEP Slide 4 2003 2009 NY State Test Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 2003 8 th Grade

What Does “Higher Standards” Mean? • More Topics? But the U. S. curriculum is

What Does “Higher Standards” Mean? • More Topics? But the U. S. curriculum is already cluttered with too many topics. 5 Slide 5 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Earlier grades? But this does not follow from the evidence. In Singapore, division of fractions: grade 6 whereas in the U. S. : grade 5 (or 4)

Lessons Learned Slide 6 6 Copyright © 2010 Pearson Education, Inc. or its affiliate(s).

Lessons Learned Slide 6 6 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • TIMSS: math performance is being compromised by a lack of focus and coherence in the “mile wide. Inch deep” curriculum • Hong Kong students outscore US students in the grade 4 TIMSS, even though Hong Kong only teaches about half the tested topics. US covers over 80% of the tested topics. • High-performing countries spend more time on mathematically central concepts: greater depth and coherence. Singapore: “Teach less, learn more. ”

Common Core State Standards Evidence Base For example: Standards from individual highperforming countries and

Common Core State Standards Evidence Base For example: Standards from individual highperforming countries and provinces were used to inform content, structure, and language. Mathematics English language arts 1. Belgium 1. Australia (Flemish) 2. Canada (Alberta) 3. China 4. Chinese Taipei 5. England 6. Finland 7. Hong Kong 8. India 9. Ireland 10. Japan 11. Korea 12. Singapore New South Wales • Victoria 2. Canada • Alberta • British Columbia • Ontario 3. England 4. Finland 5. Hong Kong 6. Ireland 7. Singapore • Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 7

Why do students have to do math problems? Slide 8 Copyright © 2010 Pearson

Why do students have to do math problems? Slide 8 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 1. To get answers because Homeland Security needs them, pronto 2. I had to, why shouldn’t they? 3. So they will listen in class 4. To learn mathematics

Answer Getting vs. Learning Mathematics United States How can I teach my kids to

Answer Getting vs. Learning Mathematics United States How can I teach my kids to get the answer to this problem? Use mathematics they already know. Easy, reliable, works with bottom half, good for classroom management. Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Japan How can I use this problem to teach mathematics they don’t already know? Slide 9 9

Three Responses to a Math Problem Slide 10 Copyright © 2010 Pearson Education, Inc.

Three Responses to a Math Problem Slide 10 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 1. Answer getting 2. Making sense of the problem situation 3. Making sense of the mathematics you can learn from working on the problem

Answer Getting Slide 11 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All

Answer Getting Slide 11 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Getting the answer one way or another and then stopping Learning a specific method for solving a specific kind of problem (100 kinds a year)

Butterfly method Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved.

Butterfly method Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 12

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 13

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 13

Use butterflies on this TIMSS item 1/2 + 1/3 +1/4 = Copyright © 2010

Use butterflies on this TIMSS item 1/2 + 1/3 +1/4 = Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 14

Foil FOIL Slide 15 15 Copyright © 2010 Pearson Education, Inc. or its affiliate(s).

Foil FOIL Slide 15 15 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • (a + b)(c +d) = ac + bc + ad + bd • Use the distributive property • This IS the distributive property when a is a sum: a(x + y) = ax + ay • Sum of products = product of sums • It works for trinomials and polynomials in general

Answers are a black hole: hard to escape the pull Slide 16 Copyright ©

Answers are a black hole: hard to escape the pull Slide 16 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Answer getting short circuits mathematics, especially making mathematical sense • High-achieving countries devise methods for slowing down, postponing answer getting

A dragonfly can fly 50 meters in 2 seconds. 17 Slide 17 Copyright ©

A dragonfly can fly 50 meters in 2 seconds. 17 Slide 17 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. What question can we ask?

Rate × Time = Distance Copyright © 2010 Pearson Education, Inc. or its affiliate(s).

Rate × Time = Distance Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 18 Slide 18

Posing the problem Slide 19 Copyright © 2010 Pearson Education, Inc. or its affiliate(s).

Posing the problem Slide 19 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Whole class: pose problem, make sure students understand the language, no hints at solution • Focus students on the problem situation, not the question/answer game. Hide question and ask them to formulate questions that make the situation into a word problem • Ask 3 -6 questions about the same problem situation; ramp questions up toward key mathematics that transfers to other problems

Bob, Jim and Cathy each have some money. The sum of Bob's and Jim's

Bob, Jim and Cathy each have some money. The sum of Bob's and Jim's money is $18. 00. The sum of Jim's and Cathy's money is $21. 00. The sum of Bob's and Cathy's money is $23. 00. Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 20

What problem to use? Slide 21 Copyright © 2010 Pearson Education, Inc. or its

What problem to use? Slide 21 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Problems that draw thinking toward the mathematics you want to teach. NOT too routine, right after learning how to solve • Ask about a chapter: what is the most important mathematics students should take with them? Find problems that draw attention to this math • Near end of chapter, external problems needed, e. g. Shell Centre

What do we mean by conceptual coherence? Apply one important concept in 100 situations

What do we mean by conceptual coherence? Apply one important concept in 100 situations rather than memorizing 100 procedures that do not transfer to other situations: Slide 22 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. – Typical practice is to opt for short-term efficiencies, rather than teach for general application throughout mathematics. – Result: typical students do OK on unit tests, but don’t remember what they ‘learned’ later when they need to learn more mathematics – Use basic “rules of arithmetic” (same as algebra) instead of clutter of specific named methods

Teaching against the test 3 + 5 = [ ] 3 + [ ]

Teaching against the test 3 + 5 = [ ] 3 + [ ] = 8 [ ] + 5 = 8 Slide 23 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 8 - 3 = 5 8 - 5 = 3

Anna bought 3 bags of red gumballs and 5 bags of white gumballs. Each

Anna bought 3 bags of red gumballs and 5 bags of white gumballs. Each bag of gumballs had 7 pieces in it. Which expression could Anna use to find the total number of gumballs she bought? Slide 24 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. A. (7 × 3) + 5 = B. (7 × 5) + 3 = C. 7 × (5 + 3) = D. 7 + (5 × 3) =

Math Standards Mathematical Practice: varieties of expertise that math educators should seek to develop

Math Standards Mathematical Practice: varieties of expertise that math educators should seek to develop in their students. Mathematical Content: Mathematical Understanding: what kids need to understand. Slide 25 25 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Mathematical Performance: what kids should be able to do.

Standards for Mathematical Content Organization by Grade Bands and Domains K– 5 6– 8

Standards for Mathematical Content Organization by Grade Bands and Domains K– 5 6– 8 Counting and Cardinality Ratios and Proportional Relationships Operations and Algebraic Thinking The Number System Expressions and Equations Number and Operations —Fractions Geometry Measurement and Data Statistics and Probability Number and Quantity Algebra Functions Modeling Geometry Statistics and Probability Functions Geometry (Common Core State Standards Initiative 2010) Slide 26 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Number and Operations in Base Ten High School

Progressions within and across Domains Copyright © 2010 Pearson Education, Inc. or its affiliate(s).

Progressions within and across Domains Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 27 Daro, 2010 2 7

Math Content Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved.

Math Content Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Greater focus – in elementary school, on whole number operations and the quantities they measure, specifically: Grades K-2 Addition and subtraction Grades 3 -5 Multiplication and division and manipulation and understanding of fractions (best predictor algebraic performance) Grades 6 -8 Proportional reasoning, geometric measurement and introducing expressions, equations, linear algebra Slide 28

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 29 Slide

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 29 Slide 29

Why begin with unit fractions? Copyright © 2010 Pearson Education, Inc. or its affiliate(s).

Why begin with unit fractions? Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 30 Slide 30

Unit Fractions 31 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights

Unit Fractions 31 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 31

Units are things that you count Daro, 2010 32 Slide 32 Copyright © 2010

Units are things that you count Daro, 2010 32 Slide 32 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Objects • Groups of objects • 100 • ¼ unit fractions • Numbers represented as expressions

Units add up 3 apples + 5 apples = 8 apples 3 ones +

Units add up 3 apples + 5 apples = 8 apples 3 ones + 5 ones = 8 ones 3 tens + 5 tens = 8 tens 3 inches + 5 inches = 8 inches 3 tenths + 5 tenths = 8 tenths 3(¼) + 5(¼) = 8(¼) 3(x + 1) + 5(x+1) = 8(x+1) Daro, 2010 33 Slide 33 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • •

There are 125 sheep and 5 dogs in a flock. Slide 34 Copyright ©

There are 125 sheep and 5 dogs in a flock. Slide 34 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. How old is the shepherd?

A Student’s Response There are 125 sheep and 5 dogs in a flock. How

A Student’s Response There are 125 sheep and 5 dogs in a flock. How old is the shepherd? 125 x 5 = 625 extremely big 125 - 5 = 120 still big 125 5 = 25 That works! Slide 35 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 125 + 5 = 130 too big

How CCLS support change The new standards support improved curriculum and instruction due to

How CCLS support change The new standards support improved curriculum and instruction due to increased: (Massachusetts State Education Department) Slide 36 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. – FOCUS, via critical areas at each grade level – COHERENCE, through carefully developed connections within and across grades – RIGOR, including a focus on College and Career Readiness and Standards for Mathematical Practice throughout Pre-K through 12

Critical Areas Grade level PK K 1 2 3 4 5 6 7 8

Critical Areas Grade level PK K 1 2 3 4 5 6 7 8 # of Critical Areas 2 2 4 4 4 3 3 4 4 3 Slide 37 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • There are two to four critical areas for instruction in the introduction for each grade level, model course or integrated pathway. • They bring focus to the standards at each grade by providing the big ideas that educators can use to build their curriculum and to guide instruction.

(Page 39) Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved.

(Page 39) Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 38 Slide 38

Critical Areas: Grade 6 Slide 39 Copyright © 2010 Pearson Education, Inc. or its

Critical Areas: Grade 6 Slide 39 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Ratio and Rate: • Connecting to whole number multiplication and division. • Equivalent ratios derive from, and extend, pairs of rows in the multiplication table. Number: • Dividing fractions in general. • Extending rational number system to negative integers (order, absolute value).

Critical Areas: Grade 6 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All

Critical Areas: Grade 6 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Expression and Equations: • Use of variables, equivalent expressions. • Solve simple one-step equations. Statistics: • Different ways to measure center of data. Geometry: • Find areas of shapes by decomposing, rearranging or removing pieces, and relating shapes to rectangles. Slide 40

(Page 46) Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved.

(Page 46) Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 41 Slide 41

Critical Areas: Grade 7 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All

Critical Areas: Grade 7 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Proportional Relationships: • Use to solve variety of percent problems. • Graph and understand unit rate as the steepness of the line, or slope. Unified Understanding of Number: • Fraction, decimal and percent are different representations of rational numbers. • Same properties and operations apply to negative numbers. Slide 42

Critical Areas: Grade 7 Slide 43 Copyright © 2010 Pearson Education, Inc. or its

Critical Areas: Grade 7 Slide 43 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Geometry: • Area and circumference of a circle. • Surface area of solids. • Scale drawing and informal constructions. • Relationships among plane figures Data: • Compare two data distributions, to see differences between populations. • Informal work with random sampling.

(Page 52) Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved.

(Page 52) Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 44 Slide 44

Critical Areas: Grade 8 Slide 45 Copyright © 2010 Pearson Education, Inc. or its

Critical Areas: Grade 8 Slide 45 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Proportional Reasoning: • Equations for proportions as special linear equations: y = mx or y/x = m • Constant of proportionality is the slope, graphs are lines through the origin. Functions: • Function as a rule that assigns to each input exactly one output. • Functions describe situations where one quantity determines another.

Critical Areas: Grade 8 Slide 46 Copyright © 2010 Pearson Education, Inc. or its

Critical Areas: Grade 8 Slide 46 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Geometry: Ideas about distance and angles, how they behave under translations, rotations, reflections and dilations and ideas about congruence and similarity

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 47 Slide

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 47 Slide 47

How CCLS support change The new standards support improved curriculum and instruction due to

How CCLS support change The new standards support improved curriculum and instruction due to increased: (Massachusetts State Education Department) Slide 48 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. – FOCUS, via critical areas at each grade level – COHERENCE, through carefully developed connections within and across grades – RIGOR, including a focus on College and Career Readiness and Standards for Mathematical Practice throughout Pre-K through 12

A Coherent Curriculum Slide 49 Copyright © 2010 Pearson Education, Inc. or its affiliate(s).

A Coherent Curriculum Slide 49 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Is organized around the big ideas of mathematics • Clearly shows how standards are connected within each grade • Builds concepts through logical progressions across grades that reflect the discipline itself.

International Comparison Charts in the next three slides are taken from: Schmidt, W. H.

International Comparison Charts in the next three slides are taken from: Schmidt, W. H. , Houang, R. , & Cougan, L. (2002). A coherent curriculum: The case of Mathematics. American educator, 26(2), 10 -26, 47 -48. As cited by Massachusetts State Education Department Slide 50 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. The mathematics curriculum of top-achieving countries on international assessments looks different from the U. S. in terms of topic placement

Topic Placement in Top Achieving Countries Copyright © 2010 Pearson Education, Inc. or its

Topic Placement in Top Achieving Countries Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 51

Topic Placement in the U. S. Copyright © 2010 Pearson Education, Inc. or its

Topic Placement in the U. S. Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 52

International Comparison In what ways do the curricula of the topachieving countries exhibit coherence?

International Comparison In what ways do the curricula of the topachieving countries exhibit coherence? Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 53

Domain Progression in the New Standards Copyright © 2010 Pearson Education, Inc. or its

Domain Progression in the New Standards Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 54 (Massachusetts State Education Department)

How CCLS support change The new standards support improved curriculum and instruction due to

How CCLS support change The new standards support improved curriculum and instruction due to increased: (Massachusetts State Education Department) Slide 55 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. – FOCUS, via critical areas at each grade level – COHERENCE, through carefully developed connections within and across grades – RIGOR, including a focus on College and Career Readiness and Standards for Mathematical Practice throughout Pre-K through 12

Standards for Mathematical Practice Slide 56 Copyright © 2010 Pearson Education, Inc. or its

Standards for Mathematical Practice Slide 56 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 1 Make sense of problems and persevere in solving them. 2 Reason abstractly and quantitatively. 3 Construct viable arguments and critique the reasoning of others. 4 Model with mathematics. 5 Use appropriate tools strategically. 6 Attend to precision. 7 Look for and make use of structure. 8 Look for and express regularity in repeated reasoning. 56

Take the number apart? Slide 57 Copyright © 2010 Pearson Education, Inc. or its

Take the number apart? Slide 57 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Tina, Emma and Jen discuss this expression: 6 × 5⅓ Tina: I know a way to multiply with a mixed number that is different from what we learned in class. I call my way ‘take the number apart. ’ I’ll show you. First, I multiply the 5 by the 6 and get 30. Then I multiply the ⅓ by the 6 and get 2. Finally, I add the 30 and the 2 to get my answer, which is 32. 57

Take the number apart? Slide 58 58 Copyright © 2010 Pearson Education, Inc. or

Take the number apart? Slide 58 58 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Tina: It works whenever I have to multiply a mixed number by a whole number. Emma: Sorry Tina, but that answer is wrong! Jan: No, Tina’s answer is right for this one problem, but ‘take the number apart’ doesn’t work for other fraction problems. Which of the three girls do you think is right? Justify your answer mathematically.

Distributive Property 5⅓ = 5 + ⅓ 6 × 5⅓ = 6(5 + ⅓)=

Distributive Property 5⅓ = 5 + ⅓ 6 × 5⅓ = 6(5 + ⅓)= 6 × 5 + 6 × ⅓ Since a(b + c) = ab + ac Could illustrate with area of rectangle 6 by 5⅓ Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 59 59

NCTM process standards Slide 60 Copyright © 2010 Pearson Education, Inc. or its affiliate(s).

NCTM process standards Slide 60 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Problem Solving • Reasoning and Proof • Communication • Representation • Connections 60

National Research Council’s report Adding It Up: Slide 61 Copyright © 2010 Pearson Education,

National Research Council’s report Adding It Up: Slide 61 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Conceptual Understanding (comprehension of mathematical concepts, operations and relations) • Procedural Fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately) • Adaptive Reasoning • Strategic Competence • Productive Disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy) 61

New York State Assessment Transition Plan Revised October 20, 2011 ELA & Math DRAFT

New York State Assessment Transition Plan Revised October 20, 2011 ELA & Math DRAFT 2 The PARCC assessments are scheduled to be operational in 2014 -15 and are subject to adoption by the New York State Board of Regents. The PARCC assessments are still in development and the role of PARCC assessments as Regents assessments will be determined. All PARCC assessments will be aligned to the Common Core. 3 The names of New York State’s Mathematics Regents exams are expected to change to reflect the new alignment of these assessments to the Common Core. For additional information about the upper-level mathematics course sequence and related standards, see the “Traditional Pathway” section of Common Core Mathematics Appendix A. 4 The timeline for Regents Math roll-out is under discussion. 5 New York State is a member of the NCSC national alternate assessments consortium that is engaged in research and development of new alternate assessments for alternate achievement standards. The NCSC assessments are scheduled to be operational in 2014 -15 and are subject to adoption by the New York State Board of Regents. Slide 62 62 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 1 New ELA assessments in grades 9 and 10 will begin during the 2012 -13 school year and will be aligned to the Common Core, pending funding.

Instructional Expectations 2011 -12 Slide 63 Copyright © 2010 Pearson Education, Inc. or its

Instructional Expectations 2011 -12 Slide 63 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Strengthening student work Curriculum Assessment Classroom instruction • Strengthening teacher practice Feedback

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 64 Slide

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 64 Slide 64

Instructional Expectations 2011 -12: Core Documents Slide 65 Copyright © 2010 Pearson Education, Inc.

Instructional Expectations 2011 -12: Core Documents Slide 65 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Framework for Teaching, Charlotte Danielson Depth of Knowledge, Norman Webb Understanding by Design, Grant Wiggins Universal Design for Learning Curriculum Mapping, Heidi Hayes Jacobs

Curriculum Mapping: Heidi Hayes Jacobs A subject or course’s Essential Map is developed by

Curriculum Mapping: Heidi Hayes Jacobs A subject or course’s Essential Map is developed by identifying: Slide 66 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. The core curriculum concepts The critical focal skills Benchmark assessments Common essential questions Essential learnings / power standards 66

Specifics of Math Task: Slide 67 Copyright © 2010 Pearson Education, Inc. or its

Specifics of Math Task: Slide 67 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Will be administered to all students in spring 2012 • Based on Standards for Math Practice # 3 and 4 and selected domains • Sample tasks, curriculum and student work available at Do. E website

Spring 2012 Task – Domains Chosen Grade Number and Operations in Base Ten Operations

Spring 2012 Task – Domains Chosen Grade Number and Operations in Base Ten Operations and Algebraic Thinking Number and Operations—Fraction Ratios and Proportional Relationships Expressions and Equations Reasoning with Equations and Inequalities Congruence 68 Slide 68 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 1 -2 3 4 -5 6 -7 8 Algebra Geometry Domain

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 69 Slide

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 69 Slide 69

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 70 Slide

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 70 Slide 70

71 Slide 71 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights

71 Slide 71 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Math Tasks

Some Criteria for Choice of Tasks Slide 72 72 Copyright © 2010 Pearson Education,

Some Criteria for Choice of Tasks Slide 72 72 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Level of challenge: accessible to the struggling, challenging enough for the advanced • Multiple points of entry • Various solution pathways • Identifying the math concept involved with, and strengthened by, working on the task

How many 'C' balls does it take to balance one 'A' ball? Copyright ©

How many 'C' balls does it take to balance one 'A' ball? Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 73

Some Criteria for Choice of Tasks Slide 74 74 Copyright © 2010 Pearson Education,

Some Criteria for Choice of Tasks Slide 74 74 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Opportunities to exercise the standards for mathematical practice • Opportunities to bring out student misconceptions, which can be identified and addressed

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 75 Slide

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 75 Slide 75

Research on Retention of Learning: Shell Center: Swan et al Copyright © 2010 Pearson

Research on Retention of Learning: Shell Center: Swan et al Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 76

Pedagogy Slide 77 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights

Pedagogy Slide 77 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Make conceptions and misconceptions visible to the student • Students need to be listened to and responded to • Partner work • Revise conceptions • Debug processes • Meta-cognitive skills

A Problem (DO NOT SOLVE) What questions do you ask yourself as you encounter

A Problem (DO NOT SOLVE) What questions do you ask yourself as you encounter this problem? How do these questions help you to develop a solution approach? Slide 78 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Make as many rectangles as you can with an area of 24 square units. Use only whole numbers for the length and width. Sketch the rectangles, and write the dimensions on the diagrams. Write the perimeter of each one next to the sketch.

Meta-Cognition • Thinking about thinking. • The unconscious process of cognition. • Meaning-making Slide

Meta-Cognition • Thinking about thinking. • The unconscious process of cognition. • Meaning-making Slide 79 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. It is hard to articulate how you think about thinking. It is even harder to model

Meta-cognition implications for lessons. • • • Slide 80 Copyright © 2010 Pearson Education,

Meta-cognition implications for lessons. • • • Slide 80 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Make thinking public Use multiple representations Offer different approaches to solution Ask questions about the problem posed. Set a context, define the why of the problem • Focus students on their thinking, not the solution • Solve problems with partners • Prepare to present strategies

You were supposed to add A and B. By accident, you subtracted B from

You were supposed to add A and B. By accident, you subtracted B from A and got 4. This number is different from the correct answer by 12. What is A? Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 81

Slide 82 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved.

Slide 82 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Aligning Tasks to the Common Core Learning Standards

Comparing Two Mathematical Tasks TASK A MAKING CONJECTURES Complete the conjecture based on the

Comparing Two Mathematical Tasks TASK A MAKING CONJECTURES Complete the conjecture based on the pattern you observe in the specific cases. 29. Conjecture: The sum of any two odd numbers is ______? 1 + 1 = 2 7 + 11 = 18 1 + 3 = 4 13 + 19 = 32 3 + 5 = 8 201 + 305 = 506 Slide 83 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 30. Conjecture: The product of any two odd numbers is ____? 1 x 1 = 1 7 x 11 = 77 1 x 3 = 3 13 x 19 = 247 3 x 5 = 15 201 x 305 = 61, 305

Comparing Two Mathematical Tasks Slide 84 Copyright © 2010 Pearson Education, Inc. or its

Comparing Two Mathematical Tasks Slide 84 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. TASK B MAKING CONJECTURES Complete the conjecture based on the pattern you observe in the specific cases. Then explain why the conjecture is always true or show a case in which it is not true. 29. Conjecture: The sum of any two odd numbers is ______? 1 + 1 = 2 7 + 11 = 18 1 + 3 = 4 13 + 19 = 32 3 + 5 = 8 201 + 305 = 506 30. Conjecture: The product of any two odd numbers is ____? 1 x 1 = 1 7 x 11 = 77 1 x 3 = 3 13 x 19 = 247 3 x 5 = 15 201 x 305 = 61, 305

Draw a Picture Every odd number (like 11 and 13) has one loner number.

Draw a Picture Every odd number (like 11 and 13) has one loner number. Add the two loner numbers and you will get an even number (24). Now add all together the loner numbers and the other two (now even) numbers. Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 85

Build a Model If I take the numbers 5 and 11 and organize the

Build a Model If I take the numbers 5 and 11 and organize the counters as shown, you can see the pattern. Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. You can see that when you put the sets together (add the numbers), the two extra blocks will form a pair and the answer is always even. This is because any odd number will have an extra block and the two extra blocks for any set of two odd numbers will always form a pair. Slide 86

Logical Argument An odd number = [an] even number + 1. e. g. 9

Logical Argument An odd number = [an] even number + 1. e. g. 9 = 8 + 1 Therefore as an odd number = an even number + 1, if you add two of them together, you get an even number + 2, which is still an even number. Slide 87 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. So when you add two odd numbers you are adding an even no. + 1 + 1. So you get an even number. This is because it has already been proved that an even number + an even number = an even number.

Use Algebra If a and b are odd integers, then a and b can

Use Algebra If a and b are odd integers, then a and b can be written a = 2 m + 1 and b = 2 n + 1, where m and n are other integers. If a = 2 m + 1 and b = 2 n + 1, then a + b = 2 m + 2 n + 2. If a + b = 2(m + n + 1), then a + b is an even integer. Slide 88 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. If a + b = 2 m + 2 n + 2, then a + b = 2(m + n + 1).

Comparing Two Mathematical Tasks How are the two versions of the task the same

Comparing Two Mathematical Tasks How are the two versions of the task the same and how are they different? Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 89

Tasks A and B Different l Task B asks students to develop an argument

Tasks A and B Different l Task B asks students to develop an argument that explains why the conjecture is always true (or not) l Task A can be completed with limited effort; Task B requires considerable effort – students need to figure out WHY this conjecture holds up l The amount of thinking and reasoning required l The number of ways the problem can be solved l The range of ways to enter the problem Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Same l Both ask students to complete a conjecture about odd numbers based on a set of finite examples that are provided Slide 90

Standards for Mathematical Practice Common Core State Standards for Mathematics, 2010, pp. 6 -7

Standards for Mathematical Practice Common Core State Standards for Mathematics, 2010, pp. 6 -7 Slide 91 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

Standards for Mathematical Practice Common Core State Standards for Mathematics, 2010, pp. 6 -7

Standards for Mathematical Practice Common Core State Standards for Mathematics, 2010, pp. 6 -7 Slide 92 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

What is the value of 17 × 13 + 61 × 13 + 22

What is the value of 17 × 13 + 61 × 13 + 22 × 13? Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 93 Slide 93

Analysis of Tasks Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights

Analysis of Tasks Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 94 Slide 94

Levels of Cognitive Demand Lower-level Slide 95 Copyright © 2010 Pearson Education, Inc. or

Levels of Cognitive Demand Lower-level Slide 95 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Memorization • Procedures without connections Higher-level • Procedures with connections • Doing mathematics 95

Norman Webb’s Depth of Knowledge Slide 96 96 Copyright © 2010 Pearson Education, Inc.

Norman Webb’s Depth of Knowledge Slide 96 96 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Level 1: Recall and Reproduction • Level 2: Skills and Concepts • Level 3: Strategic Thinking • Level 4: Extended Thinking

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 97 Slide

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 97 Slide 97

Cognitive Complexity BLOOMS TAXONOMY WEBB’S DEPTH OF KNOWLEDGE- the recall of specifics and universals,

Cognitive Complexity BLOOMS TAXONOMY WEBB’S DEPTH OF KNOWLEDGE- the recall of specifics and universals, involving more than bringing to mind the appropriate material RECALL- recall of fact, information, or procedure COMPREHENSION- Ability to process knowledge on a low level such that the knowledge can be reproduced or communicated without a verbatim repetition ANALYSIS- the breakdown of a situation into its component parts STRATEGIC THINKING- requires reasoning, developing a plan or sequence of steps; has some complexity; more than one possible answer SYNTHESIS & EVALUATION- putting together elements and parts to form a while, then making value judgments about the method EXTENDED THINKING- requires an investigation; time to think and process multiple conditions of the problem / task; non-routine manipulations Slide 98 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. APPLICATION- the use of abstractions in concrete APPLICATION of SKILL / CONCEPT- use of situations information, conceptual knowledge, procedures of two or more steps

What is Depth of Knowledge? • A language system used to describe different levels

What is Depth of Knowledge? • A language system used to describe different levels of complexity • A framework for evaluating curriculum, objectives, and assessments so they can be studied for alignment Slide 99 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Focuses on content and cognitive demand of test items, instructional strategies, and performance objectives

DOK Levels Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved.

DOK Levels Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Level 1 measures Recall at a literal level. Level 2 measures a Skill or Concept at an interpretive level. Level 3 measures Strategic Thinking at an evaluative level. Level 4 measures Extended Thinking and Reasoning Slide 100

DOK Level 1: Mathematics • Recall and recognize information such as facts, definitions, theorems,

DOK Level 1: Mathematics • Recall and recognize information such as facts, definitions, theorems, terms, formulas or procedures • Demonstrate an understanding of fundamental math concepts Slide 101 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Solve one-step problems, apply formulas, and perform well-defined algorithms

DOK 1 What is the place value of 9 in the number 74. 295?

DOK 1 What is the place value of 9 in the number 74. 295? A. hundreds B. tenths Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. C. hundredths D. thousandths Slide 102

DOK Level 2: Mathematics Slide 103 Copyright © 2010 Pearson Education, Inc. or its

DOK Level 2: Mathematics Slide 103 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. The cognitive demands are more complex than in Level 1. Engage in mental processing beyond recall or habitual response: • Determine how to approach a problem • Solve routine multi-step problems • Estimate quantities, amounts, etc. • Use and manipulate multiple formulas, definitions, theorems, or a combination of these • Collect, organize, classify, display, and compare data • Extend a pattern

DOK 2 Draw the next figure in the following pattern: Copyright © 2010 Pearson

DOK 2 Draw the next figure in the following pattern: Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 104

DOK 2 On a road trip from Georgia to Slide 105 Copyright © 2010

DOK 2 On a road trip from Georgia to Slide 105 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Oklahoma, Maria determined that she would cover about 918 miles. What speed would she need to average to complete the trip in no more than 15 hours of driving time?

DOK Level 3: Mathematics Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All

DOK Level 3: Mathematics Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Engage in abstract, complex thinking • Determine which concepts to use in solving complex problems • Use multiple concepts to solve a problem • Reason, plan, and use evidence to explain and justify thinking • Make conjectures • Interpret information from complex graphs • Draw conclusions from logical arguments Slide 106

DOK 3 Find the next three items in the pattern and give the rule

DOK 3 Find the next three items in the pattern and give the rule for following the pattern of numbers: Slide 107 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 1, 4, 3, 6, 5, 8, 7, 10…

DOK 3 A local bakery celebrated its one year anniversary on Saturday. On that

DOK 3 A local bakery celebrated its one year anniversary on Saturday. On that day, every 4 th customer received a free cookie. Every 6 th customer received a free muffin. A. Did the 30 th customer receive a free cookie, free muffin, both, or neither? Show or explain how you got your answer. B. Casey was the first customer to receive both a free cookie and a free muffin. What number customer was Casey? Show or explain how you got your answer. D. On that day the bakery gave away a total of 29 free cookies. What was the total number of free muffins the bakery gave away on that day? Show or explain how you got your answer. Slide 108 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. C. Tom entered the bakery after Casey. He received a free cookie only. What number customer could Tom have been? Show or explain how you got your answer.

DOK Level 4: Mathematics Extended Thinking/Reasoning requires complex reasoning, planning, developing, and thinking most

DOK Level 4: Mathematics Extended Thinking/Reasoning requires complex reasoning, planning, developing, and thinking most likely over an extended period of time. Slide 109 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. The extended time period is not a distinguishing factor if the required work is only repetitive and does not require applying significant conceptual understanding and higher-order thinking.

DOK 4 Slide 110 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All

DOK 4 Slide 110 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Take the bakery problem and ask students to write equations. Solve the system of equations and explain why it satisfies the conditions. Determine what any customer might receive, i. e. the 1000 th customer. Refer back to the task on equations in the form y=mx + b.

DOK 4 Slide 111 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All

DOK 4 Slide 111 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. George Smith charges $4. 00 an hour for his services to walk/feed/water outdoor pets when his clients take weekend trips. Charles Wood charges $45. 00 for weekly lawn care – mowing, weeding, raking. Marty Rogers cleans and organizes items in sheds/garages at the rate of $6. 50 per hour. If each of these boys’ families needs the services of the other two boys, determine a fair way (as fair as possible) to arrange for services to be rendered among the three families without the exchange of money.

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 112

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 112

Implementation of Tasks Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights

Implementation of Tasks Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 113

Standards for Mathematical Practice “Not all tasks are created equal, and different tasks will

Standards for Mathematical Practice “Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking. ” Stein, Smith, Henningsen, & Silver, 2000 1 1 Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997 Slide 114 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. “The level and kind of thinking in which students engage determines what they will learn. ”

Standards for Mathematical Practice 1 1 Slide 115 Copyright © 2010 Pearson Education, Inc.

Standards for Mathematical Practice 1 1 Slide 115 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. The eight Standards for Mathematical Practice – place an emphasis on student demonstrations of learning… Equity begins with an understanding of how the selection of tasks, the assessment of tasks, the student learning environment create great inequity in our schools…

Opportunities for all students to engage in challenging tasks? • Examine tasks in your

Opportunities for all students to engage in challenging tasks? • Examine tasks in your instructional 1 Slide 116 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. materials: – Higher cognitive demand? – Lower cognitive demand? • Where are the challenging tasks? • Do all students have the opportunity to grapple with challenging tasks? • Examine the tasks in your assessments: – Higher cognitive demand? 1 – Lower cognitive demand?

The nature of tasks used in the classroom… will impact student learning! Copyright ©

The nature of tasks used in the classroom… will impact student learning! Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 1 1 Slide 117

But WHAT TEACHERS DO with the tasks matters too! The Mathematical Tasks Framework Copyright

But WHAT TEACHERS DO with the tasks matters too! The Mathematical Tasks Framework Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 1 1 Slide 118

Students’ beliefs about their intelligence • Fixed mindset: Dweck, 2007 1 1 Slide 119

Students’ beliefs about their intelligence • Fixed mindset: Dweck, 2007 1 1 Slide 119 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. – Avoid learning situations if they might make mistakes – Try to hide, rather than fix, mistakes or deficiencies – Decrease effort when confronted with challenge • Growth mindset: – Work to correct mistakes and deficiencies – View effort as positive; increase effort when challenged

Students can develop growth mindsets • Explicit instruction about the brain, its NCSM Position

Students can develop growth mindsets • Explicit instruction about the brain, its NCSM Position Paper #7 Promoting Positive Self-Beliefs 1 2 Slide 120 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. function, and that intellectual development is the result of effort and learning has increased students’ achievement in middle school mathematics. • Teacher praise influences mindsets – Fixed: Praise refers to intelligence – Growth: Praise refers to effort, engagement, perseverance

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 121 Slide

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 121 Slide 121

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 122 Slide

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 122 Slide 122

Changing View of Assessment: 123 Slide 123 Copyright © 2010 Pearson Education, Inc. or

Changing View of Assessment: 123 Slide 123 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Assessment for Learning

Formative Assessment Strategies Slide 124 Copyright © 2010 Pearson Education, Inc. or its affiliate(s).

Formative Assessment Strategies Slide 124 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 1. Clarifying, sharing and understanding goals for learning and criteria for success with learners 2. Engineering effective classroom discussions, questions, activities and tasks that elicit evidence of students’ learning 3. Providing feedback that moves learning forward 4. Activating students as owners of their own learning 5. Activating students as learning resources for one another

Slide 125 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved.

Slide 125 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. When teachers start from what it is they want students to know and design their instruction backward from that goal, then instruction is far more likely to be effective.

Two-Stage Process: • Clarifying the learning goals • Establishing success criteria Slide 126 Copyright

Two-Stage Process: • Clarifying the learning goals • Establishing success criteria Slide 126 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. “…discrepancies in beliefs about what it is that counts as learning in mathematics classrooms may be a significant factor in the achievement gaps observed…”

Ambiguities inherent in mathematics Students who do not understand what is important and what

Ambiguities inherent in mathematics Students who do not understand what is important and what is not important will be at a very real disadvantage Slide 127 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 6½ 6 x 61

Eliciting evidence of student learning Slide 128 Copyright © 2010 Pearson Education, Inc. or

Eliciting evidence of student learning Slide 128 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • By crafting questions that explicitly build in the undergeneralizations and overgeneralizations that students are known to make • The teacher is able to address students’ confusion during the lesson

Feedback that moves the learner forward Slide 129 Copyright © 2010 Pearson Education, Inc.

Feedback that moves the learner forward Slide 129 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Feedback is usually “egoinvolving” • Grades with comments are no more effective than grades alone, and much less effective than comments alone

Finding errors for themselves Slide 130 Copyright © 2010 Pearson Education, Inc. or its

Finding errors for themselves Slide 130 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • There are five answers here that are incorrect. Find them and fix them. • The answer to this question is __ Can you find a way to work it out?

Identify where students might use and extend their existing knowledge • You’ve used substitution

Identify where students might use and extend their existing knowledge • You’ve used substitution to solve all these simultaneous equations. Can you use elimination? Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 131

Encourage pupils to reflect Slide 132 Copyright © 2010 Pearson Education, Inc. or its

Encourage pupils to reflect Slide 132 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • You used two different methods to solve these problems. What are the advantages and disadvantages of each? • You have answered ___ well. Can you make up your own more difficult problems?

Have students discuss their ideas with others Slide 133 Copyright © 2010 Pearson Education,

Have students discuss their ideas with others Slide 133 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • You seem to be confusing sine and cosine. Talk to Katie about how to work out the difference. • Compare your work with Ali and write some advice to another student tackling this topic for the first time.

Activating students as owners of their own learning Slide 134 Copyright © 2010 Pearson

Activating students as owners of their own learning Slide 134 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Student motivation and engagement: cost vs. benefits • “It’s better to be thought lazy than dumb. ” • Focus on personal growth rather than a comparison with others • Green, yellow red “traffic lights”

Activating students as learning resources for one another Slide 135 Copyright © 2010 Pearson

Activating students as learning resources for one another Slide 135 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. • Group goals, so that students are working as a group, not just in a group • Individual accountability • Feedback from a peer: two stars and a wish • Internalize the learning intentions and success criteria in the context of someone else’s work

Assessment for Learning Dylan Wiliam Slide 136 Copyright © 2010 Pearson Education, Inc. or

Assessment for Learning Dylan Wiliam Slide 136 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Teachers use assessment, minute-by-minute and dayby-day, to adjust their instruction to meet their students’ learning needs.

Assessment for Learning Slide 137 Copyright © 2010 Pearson Education, Inc. or its affiliate(s).

Assessment for Learning Slide 137 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Change of focus from what the teacher is putting into the lesson, to what the learner is getting out of it. 137

Danielson’s Framework for Teaching Components of Professional Practice Slide 138 Copyright © 2010 Pearson

Danielson’s Framework for Teaching Components of Professional Practice Slide 138 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Domain 1: Planning and Preparation Component 1 a: Demonstrating Knowledge of Content and Pedagogy Component 1 b: Demonstrating Knowledge of Students Component 1 c: Selecting Instructional Goals Component 1 e: Designing Coherent Instruction Component 1 f: Assessing Student Learning

Danielson’s Framework for Teaching Components of Professional Practice Slide 139 Copyright © 2010 Pearson

Danielson’s Framework for Teaching Components of Professional Practice Slide 139 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Domain 2: The Classroom Environment Component 2 a: Creating an Environment of Respect and Rapport Component 2 b: Establishing a Culture for Learning Component 2 c: Managing Classroom Procedures Component 2 d: Managing Student Behavior Domain 3: Instruction Component 3 a: Communicating Clearly and Accurately Component 3 b: Using Questioning and Discussion Techniques Component 3 c: Engaging Students in Learning Component 3 d: Providing Feedback to Students Component 3 e: Demonstrating Flexibility and Responsiveness

Slide 140 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved.

Slide 140 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Sample Tasks

Animals Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide

Animals Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 141

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 142

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 142

Giantburgers Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide

Giantburgers Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 143

Security Camera Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved.

Security Camera Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 144

Pythagorean Triples Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved.

Pythagorean Triples Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 145

Multiplying Cells Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved.

Multiplying Cells Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 146

Circle Pattern Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved.

Circle Pattern Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 147

Skeleton Tower Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved.

Skeleton Tower Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 148

Resources Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 149

Resources Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. 149 Slide 149

Web Links 15 0 Slide 150 Copyright © 2010 Pearson Education, Inc. or its

Web Links 15 0 Slide 150 Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Common Core State Standards: http: //www. corestandards. org/ Common Core Tools: http: //commoncoretools. wordpress. com/ New York State Site for Teaching and Learning Resources http: //www. engageny. org/ PARCC: http: //www. parcconline. org/ Inside Mathematics: http: //www. insidemathematics. org/index. php/home Mathematics Assessment Project: http: //map. mathshell. org/materials/index. php Common Core Library: http: //schools. nyc. gov/Academics/Common. Core. Library/default. htm Mathematical Olympiads for Elementary and Middle Schools: www. moems. org

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 151

Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Slide 151 1 5

Pearson Professional Development pearsonpd. com Copyright © 2010 Pearson Education, Inc. or its affiliate(s).

Pearson Professional Development pearsonpd. com Copyright © 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. RONALD SCHWARZ, facilitator Ronald. schwarz@yahoo. com Slide 152