lecture eight
DESCRIPTION
LECTURE EIGHT. NATIONAL Pathway for Common Core Implementation. WELCOME. Your Faculty Kevin Baird [email protected]. Today’s Webinar. Lecture Eight Standards for Mathematical Practices In Depth. Today’s Overview. Implications from ELA: CCR Number One Knowledge: - PowerPoint PPT PresentationTRANSCRIPT
LECTURE EIGHT
NATIONALPATHWAY
FOR COMMON COREIMPLEMENTATION
Implications from ELA:CCR Number One
Knowledge: Math Practices ReviewGrouping the Standards for Math PracticesLearning Progressions Research Basis
Resources:Deconstructed StandardsMath Fluency StandardsHandouts
TODAY’S OVERVIEW
Readings: A few more introduced today
Spend time with the ProgressionsSpend time with the actual CCSS (Math is more focused)NOTES Today: “Big Rocks”
SCOPE & SEQUENCE
AN EYE ON SKILL BUILDING:
WHAT TEACHERS CAN DO..
SAFETY IN ASKING FOR HELP
A Little Review
More than Reading,Mathematics is often
Curriculum Driven
Curriculum Materials are unlikely to help (yet)
or make up for lack of math skills
MATHEMATICS
Vision Skills+ + Incentives + Resources + Action Plan = SustainableChange
Skills + Incentives + Resources + Action Plan = Confusion
Vision + Incentives + Resources + Action Plan = Anxiety
Vision Skills+ + Resources + Action Plan = Resistance
Vision Skills+ + Incentives + Action Plan = Frustration
Vision Skills+ + Incentives + Resources = Treadmill
Curriculum Mapping Implementation
Key Questions:Resources -- "Do we have tools, time, and training to map effectively?"Action Plan -- "Over the next three years, do we have attainabletimelines and goals? Who will be the responsible parties forimplementations, monitoring, and feedback?"
Vision -- "Why are we doing this?"Skills -- "How do we build effective maps?"Incentives -- "How will mapping improveteaching and learning?"
Conditions for Successful Implementation
Plan
Plan
Plan
Plan
Plan
Vision: The “Why are we doing this?” to combat confusion.Skills: The skill sets needed to combat anxiety.Incentives: Reasons, perks, advantages to combat resistanceResources: Tools and time needed to combat frustration.
Plan: Provides the direction to eliminate the treadmill effect.
Knoster, T., Villa, R., & Thousand, J. (2000)
College & Career Readiness Benchmarks The Assessment Today Middle Grades
KEY DATA REVIEW
COLLEGE READINESS BENCHMARKS BY SUBJECT
Percent of ACT-Tested School GraduatesMeeting College Readiness BenchmarksBy Subject 2011
66% of all ACT-tested high school graduates met the English CollegeReadiness Benchmark in 2011. Just1 in 4 (25%) met all four CollegeReadiness Benchmarks.
In 2011, 52% of graduates met The Reading Benchmark, while 45% met the Mathematics Benchmark. Just under 1 in 3 (30%)Met the College Readiness Benchmark in Science.
SecondFirst ThirdKindergartenNumber Sense
OperationsMeasurement
Consumer ApplicationsBasic Algebra
Advanced AlgebraGeometric ConceptsAdvanced Geometry
Data DisplaysStatistics
ProbabilityAnalysis
TrigonometrySpecial Topics
FunctionsInstructional Technology
I. Memorize Facts, Definitions, FormulasII. Perform ProceduresIII. Demonstrate UnderstandingIV. Conjecture, Analyze, Generalize, ProveV. Solve Non-Routine Problems/Make
Connections
K – 8 DOMAINS
04/21/23 • page 18
Domains K 1 2 3 4 5 6 7 8
Counting and Cardinality
Operations and Algebraic Thinking
Number and Operations in Base Ten
Measurement and Data
Geometry
Number and Operations - Fractions
Ratios and Proportional Relationships
The Number System
Expressions and Equations
Statistics and Probability
Functions
Learning Progression
Draw a basic number line from 0 to 10
Locate simple whole
numbers on a number line
Place halves in fraction form on a number
line
Locate tenths in decimal form
on a number line
Indicate the approximate location of
thirds, fourths, and fifths on a number line
Identify and locate the
approximate location of decimals in
hundredths on a number line
Compare fractions,
decimals and mixed numbers by identifying their relative position on a number line
Standard: Identify the relative position of simple positive fractions, positive mixed numbers, and positive decimals and be able to evaluate the values based on their position on a number line.
COURSES MATTER
Required FluencyK Add/subtract within 5 1 Add/subtract within 10 2 Add/subtract within 20
Add/subtract within 100 (pencil and paper) 3 Multiply/divide within 100
Add/subtract within 1,000 4 Add/subtract within 1,000,000 5 Multidigit multiplication 6 Multidigit division
Multidigit decimal operations 7 Solve px + q = r, p(x + q) = r
Grade 8: Critical AreasIn Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.
PROFICIENCY
FLUENCY
ELA
StanzaPreferencePunctuationCollaborateIllustratorBrainstormPunctuationNon-fiction
Math
AttributeDecomposeDecompositionCompositionHexagonDimensionalVerticesCategory
KINDERGARTEN ESSENTIAL VOCABULARY
24
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments / 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
The Process of “Figuring Out” and “Problem Solving” is
at least as important
as the solution itself
“There is no ‘right’ answer… some answers are more correct than others”
IN A NUTSHELL
Key Findings
Fall off begins late middle school, where Algebra starts The sequence of learning matters Curriculum should be streamlined and focused Combine conceptual understanding, procedural fluency, and automatic (quick, effortless) recall of facts Effort, not just talent, counts
2008 NATIONAL MATH ADVISORY PANEL
By the end of the third grade, students should be proficient in adding and subtracting whole numbers.
Two years later, they should be proficient in multiplying and dividing them.
By the end of the sixth grade, the students should have mastered the multiplication and division of fractions and decimals.
BENCHMARKS
To prepare students for algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency and problem-solving skills.
Findings closely tracked the 2006 NCTM Report
Fractions are especially troublesome for Americans, the report found. It pointed to the National Assessment of Educational Progress, standardized exams known as the nation’s report card, which found that almost half the eighth graders tested could not solve a word problem that required dividing fractions. Panel members said the failure to master fractions was for American students the greatest obstacle to learning algebra. Just as “plastics” was the catchword in the 1967 movie “The Graduate,” the catchword for math teachers today should be “fractions,” said Francis Fennell, president of the National Council of Teachers of Mathematics.
THE PROBLEM WITH FRACTIONS
By Grade 4:
BOTH: fluent at adding, subtracting, and multiplying whole numbers; apply place value; classify two-dimensional geometric figures.
COMMON CORE: Multiply fractions by whole numbers, understand multiplication algorithms through applications of place value and some operations.
COMMON CORE: HIGHER RIGOR
By 8th:
BOTH: Focus on algebraic expressions, equations, and functions; expect effective work with symbols to transform linear expressions and solve linear equations (beginning algebra).
COMMON CORE: Emphasis on geometric properties, using function to model relationships between quantities.
High School
BOTH: Very Similar Content, Different Sequence & Combination
COMMON CORE: Expects more rigor with polynomial functions and attributes
1. Children in the early primary grades should be allowed the time to develop whole number concepts and whole number operations informally with abundant concrete referents. Arabic symbols should be used for counting purposes only and always connected to concrete objects or pictorial representations. Informal practice with fraction concepts should be limited to experiences that arise naturally, like fair sharing or situations that involve money. Lamon (1999) claims that studies have shown that if children are given the time to develop their own reasoning for at least three years without being taught standard algorithms for operations with fractions and ratios, then a dramatic increase in their reasoning abilities occurred, including their proportional thinking (p. 5).
BROWN & QUINN 2006
2. Upper primary students should be given experiences that extend the whole number concept with an eye toward algebra involving an informal treatment of the field properties. These students need to be provided with experience in partitioning as a method for solving verbal problems involving fractions (Lamon, 1999; Huinker, 1998). The informal treatment of fractions should include manipulation of concrete objects and the use of pictorial representations, such as unit rectangles and number lines. Fraction notation must be developed, but formal fraction operations using teacher-taught algorithms should be postponed. Learning the subject of fractions will revolve around informal strategies for solving problems involving fractions. The objective at this level is to build a broad base of experience that will be the foundation for a progressively more formal approach to learning fractions.
BROWN & QUINN 2006
3. In middle school, the development of fraction operations as an extension of whole number operations should provide experiences that guide and encourage students to construct their own algorithms (Lappan & Bouck, 1998; Sharp, 1998). More time is needed to allow students to invent their own ways to operate on fractions rather than memorizing a procedure (Huinker, 1998). Progressively this development should lead to more formal definitions of fraction operations and algorithms that prepare students for the abstractions that arise later in the study of algebra (Wu, 2001). How fractions should be taught is inexorably linked to when the concepts are being presented and what impact the learned concepts will have on future mathematics courses
“Students who complete Algebra II are more than twice as likely to graduate from college compared to students with less mathematical preparation.”
RIGOR
.
It is important for students to master their basic math facts well enough that their recall becomes automatic, stored in their long-term memory, leaving room in their working memory to take in new math processes.
“For all content areas, practice allows students to achieve automaticity of basic skills — the fast, accurate and effortless processing of content information — which frees up working memory for more complex aspects of problem solving,”
The report also cited findings that students who depended on their native intelligence learned less than those who believed that success depended on how hard they worked.
The current “talent-driven approach to math, that either you can do it or you can’t, like playing the violin,” needed to be changed.
Math Anxiety is Real
It Accelerates in Middle Grades
It is firmly developed by High School
ADDRESS MATH ANXIETY
Graphic Organizer Model / Approach for Teacher Assistant
CUNNINGHAM / ROBERTS STUDY
McCormick, Kelly Experiencing the Power of Learning Mathematics through Writing (September 2010) Abstract. As part of the Writing Across the Curriculum movement, teachers are asked to integrate writing into their teaching of mathematics; however, this can be a difficult task given that most elementary school teachers have had little experience using writing as a tool to learn and communicate their understanding of mathematics. To give students in my mathematics content courses the valuable experience of writing mathematical explanations, writing has become an integral part of my courses for preservice teachers. The paper that follows focuses on how I support strong written explanations in my mathematics content courses for elementary school teachers.
Burns, Barbara A. Pre-Service Teachers’ Exposure to Using the History of Mathematics to Enhance Their Teaching of High School Mathematics (September 2010) Abstract. The history of mathematics is an important component in the learning of mathematics. This study examines how pre-service teachers view the role of history of mathematics in the high school curriculum. Quantitative and qualitative methods were used. Results showed significant changes in beliefs about how the history of mathematics should be integrated as well as preparedness to incorporate the history of mathematics in teaching.
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Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments / 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
MATH PRACTICE 1
Make sense of problems and persevere in solving them(May combine easily with MP 2, 4, 5, 7, 8)
Word problems involving critical math knowledge (e.g., a multiplication or division word problem in Grade 3 or addition and subtraction of fractions in Grade 4)
Problems that require careful review and thought Problems that take a long time to solve Problems that require a number of routine steps Problems in which each step leads to a more difficult problem
MATH PRACTICE 2
Reason abstractly and quantitatively(May combine easily with MP 1, 4, 7, 8)
Problems where students must compute and interpret remainders in word problems.
Kim is making candy bags. There will be 5 pieces of candy in each bag. She had 53 pieces of candy. She ate 14 pieces of candy. How many candy bags can Kim make now? Will there be any left over? If so, how much? Show your work.
MATH PRACTICE 2
Reason abstractly and quantitatively(May combine easily with MP 1, 4, 7, 8)
Contextual problems in which the student can gain insight into the problem by relating the algebraic form of an answer or intermediate step to the given context
Compare 2 rate plans with different rates and startup costs (p. 42 in PARCC MCF)
R1 = ax + b and R2 = cx + d ax + b = cx + d ax – cx = d – b x(a – c) = (d – b) x = (d – b)/(a – c)
MATH PRACTICE 3
Construct viable arguments and critique the reasoning of others(May combine easily with MP 6)
Gr. 3-HS
Basing explanations/reasoning on evidence such as diagrams, calculations, terms, etc.
Distinguishing correct explanation/reasoning from that which is flawed, and—if there is a flaw in the argument—explaining what it is.
Gr. 6- HS Testing propositions or conjectures with specific examples. Justifying or refuting propositions or conjectures.
HS
Stating logical assumptions being used. Determining conditions under which an argument does and
does not apply.
MATH PRACTICE 4
Model with Mathematics (Grade 3 – HS)(May combine easily with MP 1, 2, 5, 7, 8)
Word problems involving critical math knowledge (e.g., a multiplication or division word problem in Gr. 3 or addition and subtraction of fractions in Gr. 4)
Each hat has 8 stars on it. How many total stars are on 9 hats?
Multi-step contextual word problems in which the problem isn’t necessarily broken into sub-parts.
9 large trucks are carrying ½ ton of lumber each. 7 small trucks are carrying ¼ ton of lumber each. How many total tons are being carried by all of the trucks?
MATH PRACTICE 4
Model with Mathematics (Grade 6 – HS)(May combine easily with MP 1, 2, 5, 7, 8)
Applying math techniques to real-world situations Using estimates of known quantities in a chain of reasoning
that yields an estimate of an unknown quantity.
MATH PRACTICE 4
Model with Mathematics (Grade 4 Example)
There were 28 cookies on a plate.Five children each ate one cookie. Two children each ate 3 cookies.One child ate 5 cookies.The rest of the children each ate two cookies.Then the plate was empty.How many children ate two cookies? Show your work.
MATH PRACTICE 4
Model with Mathematics (High School)(May combine easily with MP 1, 2, 5, 7, 8)
Select from a data source, analyze the data, draw conclusions, and make an evaluation or recommendation.
The purpose of such tasks is not to provide a setting for the student to demonstrate data analysis skills such as box-and-whisker plots, etc. Rather, the purpose is for the student to draw conclusions in a realistic setting, generally using elementary techniques.
Tasks that require the execution of some or all of the modeling cycle in high school (see CCSSM pp. 72,73)
MATH PRACTICE 5
Use appropriate tools strategically(May combine easily with MP 1, 4, 7)
Using coordinates, diagrams, formulas, conversions, and other math knowledge as a tool without prompting students to use a specific tool.
Use calculators to: do messy calculations, simplify expressions, solve data problems, test conjectures, etc.
Note: MP 5 is not code word for “use a calculator” Note: If a student is not being strategic in using tools, then
the student is not meeting the standard
MATH PRACTICE 6
Attend to precision(May combine easily with MP 3)
Being clear and precise when defining variables – Not “Let G be gasoline”
Knowing when a “solution” is extraneous Presenting solutions to multi-step problems with valid
chains of reasoning, using symbols appropriately – Not “1 + 4 = 5 + 7 = 12”
MATH PRACTICE 7
Look for and make use of structure(May combine easily with MP 1, 2, 4, 5)
Mathematical and real-world problems that involve rewriting an expression for a purpose
Numerical problems that reward seeing structure to simplify calculations, such as: 357 + 17,999 + 1 or 37 x 25 x 4
Analyzing parts of geometric figures to solve problems Using auxiliary lines to help solve problems or prove
something
MATH PRACTICE 8
Look for and express regularity in repeated reasoning(May combine easily with MP 1, 2, 4)
Problems in which repeated calculations or repetitions of some sort lead to a conjecture
“Initially for most students, multi-digit division problems take time and effort, so they also require perseverance (MP.1) and looking for and expressing regularity in repeated reasoning (MP.8).”
Problems in which working repetitively with numerical examples leads to writing an equation or function that describes a situation
Use graphic organizers effectively Combination Note Taking
It is important for students to be able to fluently translate between language and models and to be able to summarize the concepts in their own words – graphic organizers can facilitate this development
IMPLEMENTING THE STANDARDS FOR MATHEMATICAL PRACTICE:
STRATEGIES FOR THE CLASSROOM
Procedures/Examples Graphic Representations
Summary
Use vocabulary games or activities several times per week Many students struggle because they do not understand our math
terminology, not because they “can’t do the math”Use vocabulary graphic organizers
IMPLEMENTING THE STANDARDS FOR MATHEMATICAL PRACTICE:
STRATEGIES FOR THE CLASSROOM
Word
Definition Characteristics
Examples Non-Examples
Ensure that all students have regular opportunities to engage in mathematics tasks that are challenging (high cognitive demand) Classrooms should incorporate regular use of multi-modal
representation, particularly students making diagrams for quantitative tasks and modifying pictures in geometric tasks
Make teaching and learning conversational Give students opportunities to talk about math The language of quantitative and spatial relationships is central to
success. Teach students to create and modify diagrams
IMPLEMENTING THE STANDARDS FOR MATHEMATICAL PRACTICE: STRATEGIES FOR THE CLASSROOM
Not “Problems of the Week”Standards for Mathematical Practice interact and overlap and several may be used together to solve a problem. They are not a checklist.
Every lesson should seek to build expertise in content and practice standards.
IMPLEMENTING THE STANDARDS FOR MATHEMATICAL PRACTICE
North Carolina Unpacking Standards http://www.ncpublicschools.org/acre/standards/common-core-
tools/ Each grade level document describes the mathematical practices
in terms of the behaviors that should be exhibited by students in that grade.
Arizona CCSS Documents http://www.azed.gov/standards-practices/mathematics-standards/ List the Standards for Mathematical Practice that would be most
appropriately integrated for each individual standard.
RESOURCES – MATH PRACTICES
PARCC Model Content Frameworks http://www.parcconline.org/parcc-content-frameworks Each grade-by-grade analysis contains examples of connections
between Math Practices and Content StandardsStandards Progressions Documents – Bill McCallum
http://ime.math.arizona.edu/progressions/ Each progression contains a discussion how the Math Practices are
associated with each domainCommon Core Look-Fors App
http://itunes.apple.com/us/app/common-core-look-fors-mathematics/id467263974?mt=8
Tool for capturing classroom observation data and guide to behaviors associated with the Math Practices
RESOURCES – MATH PRACTICES
Inside Mathematics http://insidemathematics.org/index.php/common-core-standards Classroom videos and tasks that illustrate students engaged in the
Math PracticesThe Illustrative Mathematics Project
http://illustrativemathematics.org Quality examples of tasks that are aligned to CCSS
CCSSO Curriculum Materials Analysis Tools http://www.mathedleadership.org/ccss/materials.html Tools for analyzing content and math practices in curricular materials
RESOURCES – MATH PRACTICES
Great Work!
