North Country Inservice HS Mathematics Common Core State Standards Mathematics Practice and Content...
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North Country Inservice HS Mathematics Common Core State Standards Mathematics Practice and Content Standards Day 1 Friday, October 19, 2012 Presenter:
North Country Inservice HS Mathematics Common Core State
Standards Mathematics Practice and Content Standards Day 1 Friday,
October 19, 2012 Presenter: Elaine Watson, Ed.D.
Slide 2
Introductions Share What feeds your soul personally? What is
your professional role? What feeds your soul professionally?
Slide 3
Volunteers for Breaks I need volunteers to remind me when we
need breaks! Every 20 minutes, we need a 2-minute movement break to
help our blood circulate to our brains. Every hour we need a
5-minute bathroom break.
Slide 4
Formative Assessment How familiar are you with the CCSSM?
Slide 5
Setting the Stage Dan Meyers TED Talk Math Class Needs a
Makeover Go to link: watsonmath.com North Country High School Math
Inservice October 19, 2012
Slide 6
CCSSM Equally Focuses on Standards for Mathematical Practice
Standards for Mathematical Content Same for All Grade Levels
Specific to Grade Level
Slide 7
8 Practice Standards Look at the handout SMP Lesson Alignment
Template For an electronic copy to use later, go to watsonmath.com
North Country High School Math Inservice October 19, 2012
Slide 8
Standards for Mathematical Practice Describe ways in which
student practitioners of the discipline of mathematics increasingly
ought to engage with the subject matter as they grow in
mathematical maturity
Slide 9
Standards for Mathematical Practice Provide a balanced
combination of Procedure and Understanding Shift the focus to
ensure mathematical understanding over computation skills
Slide 10
Standards for Mathematical Practice Students will be able to:
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.
Slide 11
Video of NYC High School Piloting the CCSS Watch first 5
minutes on Math See link in watsonmath.com
Slide 12
Standards for Mathematical Practice Some of the following
slides on the Practice Standards have been adapted from slides
presented in several online EdWeb Webinars in February through May
2012 discussing that focused on the Practice Standards by Sara
Delano Moore, Ph.D.
Slide 13
The 8 Standards for Mathematical Practice can be divided into 4
Categories Overarching Habits of Mind of a Mathematical Thinker (#
1 and # 6) Reasoning and Explaining (# 2 and # 3) Modeling and
Using Tools (# 4 and # 5) Seeing Structure and Generalizing (# 7
and # 8)
Slide 14
The 8 Standards for Mathematical Practice are fluidly connected
to each other. One action that a student performs, either
internally or externally, when solving a problem can take on
characteristics from several of the 8 Practice Standards.
Slide 15
Overarching Habits of Mind of a Mathematical Thinker 1.Make
sense of problems & persevere in solving them. Reason
abstractly and quantitatively. Construct viable arguments &
critique the reasoning of others. Model with mathematics. Use
appropriate tools strategically. 6. Attend to precision. Look for
& make use of structure. Look for & express regularity in
repeated reasoning.
Slide 16
Start with Good Problems Characteristics Example from
Illustrative Mathematics (F-BF.A.1.a, F-IF.B.4, F-IF.B.5 ) Context
relevant to students Incorporates rich mathematics Entry
points/solution pathways not readily apparent Mike likes to canoe.
He can paddle 150 feet per minute. He is planning a river trip that
will take him to a destination about 30,000 feet upstream (that is,
against the current). The speed of the current will work against
the speed that he can paddle. Let s be the speed of the current in
feet per minute. Write an expression for r(s), the speed at which
Mike is moving relative to the river bank, in terms of s. Mike
wants to know how long it will take him to travel the 30,000 feet
upstream. Write an expression for T(s), the time in minutes it will
take, in terms of s. What is the vertical intercept of T? What does
this point represent in terms of Mikes canoe trip? At what value of
s does the graph have a vertical asymptote? Explain why this makes
sense in the situation. For what values of s does T(s) make sense
in the context of the problem?
Slide 17
Make Sense of Problems (part I) Mathematically proficient
students Explain the meaning of the problem to themselves Look for
entry points to the solution Analyze givens, constraints,
relationships, goals
Slide 18
Mikes Canoe Trip Explain the meaning of the problem Entry
points Givens, constraints, relationships, goals Mike likes to
canoe. He can paddle 150 feet per minute. He is planning a river
trip that will take him to a destination about 30,000 feet upstream
(that is, against the current). The speed of the current will work
against the speed that he can paddle. Let s be the speed of the
current in feet per minute. Write an expression for r(s), the speed
at which Mike is moving relative to the river bank, in terms of s.
Mike wants to know how long it will take him to travel the 30,000
feet upstream. Write an expression for T(s), the time in minutes it
will take, in terms of s. What is the vertical intercept of T? What
does this point represent in terms of Mikes canoe trip? At what
value of s does the graph have a vertical asymptote? Explain why
this makes sense in the situation. For what values of s does T(s)
make sense in the context of the problem?
Slide 19
Persevere in Solving Them Mathematically proficient students.
Plan a solution pathway Consider analogous cases and alternate
forms Monitor progress and change course if necessary
Slide 20
Persevere in Solving Them It's not that I'm so smart, it's just
that I stay with problems longer. - Albert Einstein
Slide 21
Mikes Canoe Trip Possible solution pathways/strategies Consider
analogous cases & alternate forms Monitor progress and change
course if needed Mike likes to canoe. He can paddle 150 feet per
minute. He is planning a river trip that will take him to a
destination about 30,000 feet upstream (that is, against the
current). The speed of the current will work against the speed that
he can paddle. Let s be the speed of the current in feet per
minute. Write an expression for r(s), the speed at which Mike is
moving relative to the river bank, in terms of s. Mike wants to
know how long it will take him to travel the 30,000 feet upstream.
Write an expression for T(s), the time in minutes it will take, in
terms of s. What is the vertical intercept of T? What does this
point represent in terms of Mikes canoe trip? At what value of s
does the graph have a vertical asymptote? Explain why this makes
sense in the situation. For what values of s does T(s) make sense
in the context of the problem?
Slide 22
Make Sense of Problems (part II) Mathematically proficient
students Explain correspondence and search for trends Check their
answers using alternate methods Continually ask themselves, Does
this make sense? Understand the approaches of others
Slide 23
What can teachers do? Select rich mathematical tasks Connected
to rigorous mathematics content Resources for rigorous mathematical
tasks can be found on www.watsonmath,.com North Country High School
Math Inservice October 19, 2012 Illustrative Mathematics MARS Tasks
Inside Mathematics 3 Act Math Tasks Dan Meyer Andrew Stadel
Others
Slide 24
What can teachers do? Ask good questions Is that true every
time? Explain how you know. Have you found all the possibilities?
How can you be sure? Does anyone have the same answer but a
different way to explain it? Can you explain what youve done so
far? What else is there to do?
Slide 25
What can teachers do? Communicate to students the final
solution to a problem is less important than the skills they
develop during the process of finding the solution. The skills
developed in working through the process are long-lasting skills
that will serve them in other areas of life.
Slide 26
Attend to Precision In Vocabulary In Mathematical Symbols In
Computation In Measurement In Communication
Slide 27
How is the teacher ensuring that students are making sense of
problem and attending to precision? See Video: Discovering
Properties of Quadrilaterals on Watsonmath.com
Slide 28
Challenges to Precision Vocabulary Similar, adjacent
Mathematical Symbols == Computation and Measurement Accurate
computation Estimating when appropriate Appropriate units of
measure Communication Formulate explanations carefully Make
explicit use of definitions
Slide 29
Mikes Canoe Trip Vocabulary Mathematical Symbols Computation
& Measurement Communication Mike likes to canoe. He can paddle
150 feet per minute. He is planning a river trip that will take him
to a destination about 30,000 feet upstream (that is, against the
current). The speed of the current will work against the speed that
he can paddle. Let s be the speed of the current in feet per
minute. Write an expression for r(s), the speed at which Mike is
moving relative to the river bank, in terms of s. Mike wants to
know how long it will take him to travel the 30,000 feet upstream.
Write an expression for T(s), the time in minutes it will take, in
terms of s. What is the vertical intercept of T? What does this
point represent in terms of Mikes canoe trip? At what value of s
does the graph have a vertical asymptote? Explain why this makes
sense in the situation. For what values of s does T(s) make sense
in the context of the problem?
Slide 30
Reasoning and Explaining Make sense of problems & persevere
in solving them. 2. Reason abstractly and quantitatively. 3.
Construct viable arguments & critique the reasoning of others.
Model with mathematics. Use appropriate tools strategically. Attend
to precision. Look for & make use of structure. Look for &
express regularity in repeated reasoning.
Slide 31
2. Reason abstractly & quantitatively Mathematics in and
out of context Working with symbols as abstractions Quantitative
reasoning requires number sense Using properties of operations and
objects Considering the units involved Attending to the meaning of
quantities, not just computation
Slide 32
Construct viable arguments Understand and use assumptions,
definitions, and prior results Think about precision (MP6) Make
conjectures and build logical progressions to support those
conjectures Not just two column proofs in high school Analyze
situations by cases Positive values of X and negative values of X
Two-digit numbers vs three-digit numbers Recognize & use
counter-examples Maximum area problem
Slide 33
How do we help children learn how to reason and explain?
Provide rich problems where multiple pathways and solutions are
possible Celebrate multiple pathways to the same answer Monitor
students as they work to choose approaches to share with the whole
class Provide plenty of opportunities for students to talk to each
other Recognize the difference between a viable argument and
opinion Provide scaffolds for thembut not too many!
Slide 34
How do we help children learn how to reason and explain?
Provide plenty of opportunities for students to talk to each other.
Create a classroom culture in which all students feel safe to
express their thinking Make sure students recognize the difference
between a viable argument and an opinion Create a classroom culture
where its safe to critique each other in a respectful way Provide
scripts (sentence frames)for them to use such as those from
Accountable Talk (see resources on watsonmath.com)
Slide 35
Teacher Moves in Group Discussion By scaffolding students'
responses and contributions, teachers can quickly make a difference
in the level of rigor and productivity in classroom talk. Teachers
can bring everyone's attention to a key point By "marking" a
student's contribution "that's an important point By asking the
student to repeat the remarkor restating it in their own wordsand
indicating why the point is important. From ACCOUNTABLE TALK
SOURCEBOOK: FOR CLASSROOM CONVERSATION THAT WORKS Version 3.1 By
Sarah Michaels (Clark University), Mary Catherine OConnor (Boston
University), Megan Williams Hall (University of Pittsburgh), with
Lauren B. Resnick (University of Pittsburgh)
Slide 36
Teacher Moves in Group Discussion If someone asks a
thought-provoking question, the teacher might turn the question
back to the group Good question, what do you think? as a way to
encourage students to push their own thinking. By citing facts and
posing counterexamples, teachers can challenge students to
elaborate or clarify their arguments "but what about...? From
ACCOUNTABLE TALK SOURCEBOOK: FOR CLASSROOM CONVERSATION THAT WORKS
Version 3.1 By Sarah Michaels (Clark University), Mary Catherine
OConnor (Boston University), Megan Williams Hall (University of
Pittsburgh), with Lauren B. Resnick (University of Pittsburgh)
Slide 37 b, a = b, a < b">
Using Properties to See Structure Properties of Operations
Commutative Property of multiplication and addition Distributive
Property of multiplication over addition Identity Property of
multiplication and addition Properties of Equality Transitive
Property (if a=b and b=c, then a=c) Properties of Inequality
Exactly one of the following is true: a > b, a = b, a <
b
Slide 69
Van Hiele Levels of Geometric Thinking shows how students
increasing see more structure in shapes as they mature
mathematically Level 0 (Pre-recognition) Students do not yet see
shapes clearly enough to compare with prototypes Level 1
(Visualization) Students understand shapes by comparing to
prototypes Students do not see properties Students make decisions
based on perception, not reasoning Level 2 (Analysis) Students see
shapes as collections of properties Students do not identify
necessary and sufficient properties
Slide 70
Van Hiele Levels (cont) Level 3 (Abstraction) Students see
relationships among figures and properties Students can create
meaningful definitions and reason informally Level 4 (Deduction)
Students can construct proofs Students understand necessary &
sufficient conditions Level 5 (Rigor) Students can understand
non-Euclidean systems Students can use indirect proof and formal
deduction
Slide 71
Look for and express regularity in repeated reasoning Focus on
computation here 1 3 = Examining points on a line and slope (1,2),
m=3 (y-2)/(y-1) = 3 Attending to intermediate results
Slide 72
Look for and express regularity in repeated reasoning These
practices are about seeing the underlying mathematical principles
and generalizations. These practices have more subtlety, and are
often hard to distinguish between each other.
Slide 73
Slide 74
Pyramid of Pennies Work through Dan Meyers Pyramid of Pennies
Problem See link on watsonmath.comwatsonmath.com Use your SMP
Lesson Planning Template and fill out what Math Practices you used
when solving the problem.
Slide 75
Slide 76
Content Standard Activity Work with a partner and on your own
laptop, go to Illustrative Mathematics on watsonmath.com Go to the
HS Standards Check out the illustrations. Which SMPs would they
reinforce? Which illustrations would you use in your classroom?
Check out the modeling standards (those with a star) Why do you
think these standards were chosen as modeling standards?
Slide 77
Last But Not Least Formative Assessment Watch the two videos on
watsonmath.com:watsonmath.com My Favorite No Daily Assessment with
Tiered Exit Cards
Slide 78
Resources For resources used and/or discussed in this
presentation, go to: www.watsonmath.com North Country High School
Math Inservice October 19, 2012 Check out other resources available
on watsonmath.com: archives of past posts resource links in the
right hand column Contact Elaine at
[email protected]@gmail.com