Ohio’s New Learning Standards in Mathematics: Supporting Depth of Understanding Catherine Puster, NPESC Curriculum Karen Witt, Curriculum Director Genoa Andrea Smith, NPESC Assistant Superintendent
Ohio’s New Learning Standards in Mathematics: Supporting Depth of Understanding Catherine Puster, NPESC Curriculum Karen Witt, Curriculum Director Genoa.
Dec 27, 2015
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Ohio’s New Learning Standards in Mathematics: Supporting Depth of UnderstandingCatherine Puster, NPESC Curriculum Karen Witt, Curriculum Director GenoaAndrea Smith, NPESC Assistant Superintendent
PAGE 2
Testing Talk
• PARCC practice problems• Activity (2 min)
– Like/learn, what went well– Need help/dislike, what did you not like
Why the Common Core?: How these Standards are Different
PAGE 4
• Activity: –What didn’t you have time for –What could you get rid of–What could you spend less time on–How can you build on what you
already do
PAGE 5
Why are we doing this? We have had standards.
Before Common Core State Standards we had standards, but rarely did we have standards-based instruction.
Long lists of broad, vague statements Mysterious assessments Coverage mentality Focused on teacher behaviors – “the inputs”
5
PAGE 6
What are our expectations?
Based on the beliefs that• A quality education is a key factor in providing all children
with opportunities for their future
• It is not enough to simply complete school, or receive a credential – students need critical knowledge and skills
• This is not a 12th grade or high school issue. It is an education system issue.
Quality implementation of the Common Core State Standards is a necessary condition for providing all students with the opportunities to be successful after high school.
6
PAGE 7 7
The Background of the Common Core
Initiated by the National Governors Association (NGA) and Council of Chief State School Officers (CCSSO) with the following design principles:
• Result in College and Career Readiness
• Based on solid research and practice evidence
• fewer, higher, and clearer
PAGE 9
“These standards are not intended to be new names for old ways of doing business. They are a call to take the next step.” CCSSM, page 5
PAGE 10
Principles of the CCSS
Fewer - Clearer - Higher
• Aligned to requirements for college and career readiness
• Based on evidence
• Honest about time
10
PAGE 12
Some Old Ways of Doing Business (1 of 2)
• A different topic every day
• Every topic treated as equally important
• Elementary students dipping into advanced topics at the expense of mastering fundamentals
• Infinitesimal advance in each grade; endless review
• Incoherence and illogic – bizarre associations, or lacking a thread
PAGE 13
Some Old Ways of Doing Business (2 of 2)
• Lack of rigor• Reliance on rote learning at expense of concepts
• Aversion to repetitious practice
• Severe restriction to stereotyped problems lending themselves to mnemonics or tricks
• Lack of quality applied problems and real-world contexts
• Lack of variety in what students produce
PAGE 14
Cautions: Implementing the CCSS is...
• Not about “gap analysis”
• Not about buying a text series
• Not a march through the standards
• Not about breaking apart each standard
14
Evidence Centered Design can inform a deliberate and systematic
approach to instruction that will help to ensure daily classroom work leads to all students meeting Ohio's
New Learning Standards.
Evidence-Centered Design (ECD) in the Classroom - Start with the end in mind.
Learning Targets/Objectives
Design begins with the inferences (claims) we want to make about students—should be connected clearly to our new standards - What should students be able to DO or KNOW?
Classroom Assessments Formative/Summative
In order to support claims, we must gather evidence----what can teachers point to, underline or highlight to show that students are making progress toward doing what we claim they can do?
Classroom Activities
Classroom activities (tasks) are designed to elicit specific evidence from students in support of claims.
PARCC is using ECD to create the gr 3-11 assessments.
PAGE 17
Mathematical Practices
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.
Don’t Bureaucratize
17
PAGE 18 18
The CCSS Requires Three Shifts in Mathematics1. Focus: Focus strongly
where the Standards focus.
2. Coherence: Think across grades and link to major topics within grades.
3. Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application.
https://vimeo.com/92784227
PAGE 19
Power of the Shifts
• Know them – both the what and the why
• Internalize them
• Apply them to your decisions about Time Energy Resources Assessments Conversations with parents, students, colleagues
• Continue to engage with them: www.achievethecore.org Follow @achievethecore on Twitter
19
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Mathematics: 3 shifts
1. Focus: Focus strongly where the Standards focus.
20
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Engaging with the shift: What do you think belongs in the major work of each grade?
Grade Which two of the following represent areas of major focus for the indicated grade?
K Compare numbers Use tally marks Understand meaning of addition and subtraction
1 Add and subtract within 20 Measure lengths indirectly and by iterating length units Create and extend patterns and sequences
2 Work with equal groups of objects to gain foundations for multiplication Understand place value Identify line of symmetry in two dimensional
figures
3 Multiply and divide within 100 Identify the measures of central tendency and distribution Develop understanding of fractions as numbers
4 Examine transformations on the coordinate plane
Generalize place value understanding for multi-digit whole numbers
Extend understanding of fraction equivalence and ordering
5 Understand and calculate probability of single events Understand the place value system
Apply and extend previous understandings of multiplication and division to multiply and divide fractions
6 Understand ratio concepts and use ratio reasoning to solve problems Identify and utilize rules of divisibility Apply and extend previous understandings of
arithmetic to algebraic expressions
7Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
Use properties of operations to generate equivalent expressions
Generate the prime factorization of numbers to solve problems
8 Standard form of a linear equation Define, evaluate, and compare functions Understand and apply the Pythagorean Theorem
Alg.1 Quadratic inequalities Linear and quadratic functions Creating equations to model situations
Alg.2 Exponential and logarithmic functions Polar coordinates Using functions to model situations
21
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Focus
• Move away from "mile wide, inch deep" curricula identified in TIMSS.
• Learn from international comparisons.
• Teach less, learn more.
• “Less topic coverage can be associated with higher scores on those topics covered because students have more time to master the content that is taught.”
– Ginsburg et al., 2005
22
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Shift #1: Focus Strongly Where the Standards Focus
• Significantly narrow the scope of content and deepen how time and energy is spent in the math classroom.
• Focus deeply on what is emphasized in the standards, so that students gain strong foundations.
PAGE 24
How Are CCSS Assessments Different?An Overview
Shift 1: Focus strongly where the Standards focus
From To
Cover content that is a “mile-wide and an inch-deep”
Assess fewer topics at each grade (as required by the Standards)
Give equal importance to all content
Dedicate large majority of score points to the major work of the grade
PAGE 25
K 12
Number and Operations
Measurement and Geometry
Algebra and Functions
Statistics and Probability
Traditional U.S. Approach
25
PAGE 26
Mathematics topics
intended at each grade by
at least two-thirds of A+
countries
Mathematics topics intended at each grade by at least two-thirds of 21 U.S. states
The shape of math in A+ countries
1 Schmidt, Houang, & Cogan, “A Coherent Curriculum: The Case of Mathematics.” (2002).
26
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Focusing attention within Number and Operations
Operations and Algebraic Thinking
Expressions and Equations
Algebra
→ →
Number and Operations—Base Ten →
The Number System
→
Number and Operations—Fractions
→
K 1 2 3 4 5 6 7 8 High School
PAGE 29
Focus in K–8Grade 2 Geometry Example
Traditional Approach (Grade 2 Geometry)
CCSS-Aligned Approach (2.G.A.3)
Shawn cut a rectangle along two lines of symmetry. How many equal shares will he have?
Ms. Nim gave her students a picture of a rectangle. Then she asked them to shade in one half of the rectangle. Here are three pictures:
Which ones show one half? Explain.Source: Illustrative Mathematics. https://www.illustrativemathematics.org/illustrations/827
PAGE 30
Focus in K–8Measures of Center Example
Traditional Approach (Grade 3 Mean and Median)
CCSS-Aligned Approach(6.SP.B.4, 5c))
Source: Illustrative Mathematics. https://www.illustrativemathematics.org/illustrations/877
PAGE 31
Focus in K–8 Supporting Work Reinforcing Major Work
Traditional Approach (Grade 1)
CCSS-Aligned Approach(1.MD.C.4)
PAGE 32
Traditional Approach(Grade 7)
CCSS-Aligned Approach(7.SP.C.8)
A coin is flipped three times.
Part A: Draw a tree diagram that shows all possible outcomes.
Part B: Create an organized list that shows all possible outcomes.
Source: EngageNY. http://www.engageny.org/sites/default/files/resource/attachments/math-grade-7.pdf
Focus in K–8 Supporting Work Reinforcing Major Work
𝟐𝟖
=𝒙𝟐𝟒
students expected to get 3 heads or 3 tails
2
HHHHHTHTHHTTTHHTHTTTHTTT
PAGE 33
Grade Focus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding
K–2 Addition and subtraction, measurement using whole number quantities
3–5 Multiplication and division of whole numbers and fractions
6 Ratios and proportional reasoning; early expressions and equations
7 Ratios and proportional reasoning; arithmetic of rational numbers
8 Linear algebra and linear functions
Priorities in Mathematics
33
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Focus in K–8 Major Work of the Grade The large majority of score
points on any grade-level assessment system should be devoted to the major work of the grade.
See achievethecore.org/focus for major work at other grade levels.
PAGE 35
Content Emphases by Cluster: Grade Four
• Key: Major Clusters; Supporting Clusters; Additional Clusters
35
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Group Discussion
• Shift #1: Focus strongly where the Standards focus.
• In your groups, discuss ways to respond to the following question, “Why focus? There’s so much math that students could be learning, why limit them to just a few things?”
36
PAGE 37
Mathematics: 3 shifts
1. Focus: Focus strongly where the Standards focus.
2. Coherence: Think across grades, and link to major topics with grades.
37
PAGE 38 38
Shift #2: Coherence: Think Across Grades, and Link to Major Topics Within Grades
• Carefully connect the learning within and across grades so that students can build new understanding on foundations built in previous years.
• Begin to count on solid conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning.
PAGE 39
How Are CCSS Assessments Different?An Overview
Shift 2: Coherence: think across grades, and link to major topics within grades
From To
Assessment as a checklist of individual standards
Items that connect standards, clusters, and domains (as is natural in mathematics) as well as items that assess individual standards
Each topic in each year is treated as an independent event
Consistent representations are used for mathematics across the grades, and Content connects to and builds on previous knowledge
PAGE 41 41
Coherence: Think Across Grades
• Example: Fractions
• “The coherence and sequential nature of mathematics dictate the foundational skills that are necessary for the learning of algebra. The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percents, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected.”
• Final Report of the National Mathematics Advisory Panel (2008, p. 18)
PAGE 42
Coherence Across GradesSeeing the Structure
“The Standards were not so much assembled out of topics as woven out of progressions.”
What It Means• Aligning items to grade-level expectations requires
understanding of all the standards at that grade (e.g., NBT standards often give bounds for OA items) and understanding how the standard fits into a progression with previous and future grades
Why It Matters for Assessment• The Standards were woven out of connected topics (the
progressions) and so assessments should also rely on the connections between topics (coherence across grades).
PAGE 43
•
One of several staircases to algebra designed in the OA domain.
Alignment in Context: Neighboring Grades and Progressions
43
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4.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
5.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
6.NS. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
6.NS.1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.
Grade 4
Grade 5
Grade 6
CCSS
44
Informing Grades 1-6 Mathematics Standards Development: What Can Be Learned from High-Performing Hong Kong, Singapore, and Korea? American Institutes for Research (2009, p. 13)
PAGE 45
Traditional Progressions(Perimeter and Area)
Grade 3:
Grade 4:
Grade 5:
Grade 6:
Coherence Across GradesConsistent Progressions
Write the area of the shape.
Determine the area of the shape in square units.
Find the perimeter of the figure.
Select the rectangle with an area of 24 square units and a perimeter of 20 units.
PAGE 46
CCSS-Aligned Progressions(Area and Surface Area)
3.MD.C.6:
4.MD.A.3:
5.NF.B.4b:
6.G.A, 6.RP.A.3:
Coherence Across GradesConsistent Progressions
Find the area of each colored figure.
Karl’s rectangular vegetable garden is 20 feet by 45 feet, and Makenna’s is 25 feet by 40 feet. Whose garden is larger in area? How much larger is that garden?
An aerial photo of farmland shows the dimensions of a field in fractions of a mile. Create a model to show the area, in square miles, of a field that is 3/4 mile by 1/3 mile.
Alexis needs to paint the four exterior walls of a large rectangular barn. The length of the barn is 80 feet, the width is 50 feet, and the height is 30 feet. The paint costs $28 per gallon, and each gallon covers 420 square feet. How much will it cost Alexis to paint the barn? Explain your work.
PAGE 47
Coherence: Link to major topics within grades
Example: data representation
Standard 3.MD.3
47
PAGE 48 48
Example: Geometric Measurement
3.MD, third cluster
Coherence: Link to Major Topics Within Grades
PAGE 49
Group Discussion
• Shift #2: Coherence: Think across grades, link to major topics within grades
• In your groups, discuss what coherence in the math curriculum means to you. Be sure to address both elements—coherence within the grade and coherence across grades. Cite specific examples.
49
PAGE 52
How Are CCSS Assessments Different?An Overview
Shift 3: Rigor: in major topics pursue conceptual understanding, procedural skill and fluency, andapplication with equal intensity
From To
Unbalanced emphasis on procedure or application
Assessment of all three aspects of rigor in balance
A lack of items that require conceptual understanding
Items that require students to demonstrate conceptual understanding of the mathematics, not just the procedures
Fluency items that are only routine and ordinary
Fluency items that are presented in new ways, as well as some that are routine and ordinary
Application of mathematics to routine and contrived word problems
Application of mathematics to authentic non-routine problems and real-world situations
PAGE 53
CCSSM-aligned assessments and sets of assessments will be balanced by distributing score points across the three aspects of rigor:
RigorBalanced Assessments
Procedural skill and fluency
Application
Conceptual understanding
PAGE 54
Rigor
54
• The CCSSM require a balance of: Solid conceptual understanding Procedural skill and fluency Application of skills in problem solving situations
• Pursuit of all three requires equal intensity in time, activities, and resources.
PAGE 56
Solid Conceptual Understanding
• Teach more than “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives
• Students are able to see math as more than a set of mnemonics or discrete procedures
• Conceptual understanding supports the other aspects of rigor (fluency and application)
56
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Fluency
• The standards require speed and accuracy in calculation.
• Teachers structure class time and/or homework time for students to practice core functions such as single-digit multiplication so that they are more able to understand and manipulate more complex concepts
59
PAGE 60 60
Required Fluencies in K-6
Grade Standard Required FluencyK K.OA.5 Add/subtract within 5
1 1.OA.6 Add/subtract within 10
2 2.OA.22.NBT.5
Add/subtract within 20 (know single-digit sums from memory)Add/subtract within 100
3 3.OA.73.NBT.2
Multiply/divide within 100 (know single-digit products from memory)Add/subtract within 1000
4 4.NBT.4 Add/subtract within 1,000,000
5 5.NBT.5 Multi-digit multiplication
6 6.NS.2,3 Multi-digit divisionMulti-digit decimal operations
PAGE 61
Application
• Students can use appropriate concepts and procedures for application even when not prompted to do so.
• Teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations, recognizing this means different things in K-5, 6-8, and HS.
• Teachers in content areas outside of math, particularly science, ensure that students are using grade-level-appropriate math to make meaning of and access science content.
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Engaging with the Shift: Making a True Statement
• This shift requires a balance of three discrete components in math instruction. This is not a pedagogical option, but is required by the Standards. Using grade __ as a sample, find and copy the standards which specifically set expectations for each component.
62
Rigor = ______ + ________ + _______
PAGE 63
Group Discussion
• Shift #3: Rigor: Expect fluency, deep understanding, and application
• In your groups, discuss ways to respond to one of the following comments: “These standards expect that we just teach rote memorization. Seems like a step backwards to me.” Or “I’m not going to spend time on fluency—it should just be a natural outcome of conceptual understanding.”
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Rigor Activity
• Use your assessment and work through making an assessment item “Rigorous”
• Pick an assessment item from your group to put on chart paper– Put the before and after item on chart paper– Post on walls for a gallery walk
PAGE 66
• Activity: –What didn’t you have time for –What could you get rid of–What could you spend less time on
PAGE 67
Additional Resources:1. For a growing bank of mini-assessments that can be used in the
classroom, go to www.achievethecore.org/math/mini-assessments.2. For a growing bank of tasks that can be used for instructional or
assessment purposes, go to www.achievethecore.org/math/tasks.3. For more information about the Shifts, go to
http://achievethecore.org/shifts-mathematics 4. For Illustrative Math tasks, go to www.illustrativemath.org.5. For PARCC sample items and practice tests, grades 3–11, go to
www.parcconline.org.6. For SBAC practice tests, grades 3–high school, go to
www.smarterbalanced.org.7. For more about the progressions documents, go to
http://math.arizona.edu/~ime/progressions/#products8. Textbook Alignment, University of Michigan: Textbook Navigator/Journal –
Michigan State University: Center for the Study of Curriculum9. http://www.edreports.org/ an independent non-profit, has released free,
web-based reviews of current K-8 math instructional materials.
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Thank You!• Catherine M Puster, [email protected]• Karen Witt, [email protected]• Andrea Smith, [email protected]
• Additional information from:– Char Shyrock, iteachbay.blogspot.com
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