North Country Inservice HS Mathematics

North Country InserviceHS Mathematics

Common Core State StandardsMathematics

Practice and Content StandardsDay 1

Friday, October 19, 2012Presenter: Elaine Watson, Ed.D.

IntroductionsShare • What feeds your soul personally?• What is your professional role?• What feeds your soul professionally?

Volunteers for BreaksI 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.

Formative AssessmentHow familiar are you with the CCSSM?

Setting the Stage

Dan Meyer’s TED TalkMath Class Needs a MakeoverGo to link: watsonmath.com

“North Country High School Math Inservice October 19, 2012”

CCSSM Equally Focuses on…

Standards for Mathematical

Practice

Standards for Mathematical

Content

Same for All Grade Levels

Specific to Grade Level

8 Practice StandardsLook at the handout

SMP Lesson Alignment TemplateFor an electronic copy to use later, go to

watsonmath.com “North Country High School Math Inservice

October 19, 2012”

Standards for Mathematical PracticeDescribe ways in which

student practitioners of the discipline of mathematics

increasingly ought to engage with the subject matter

as they grow in mathematical maturity

Standards for Mathematical PracticeProvide a balanced combination of

Procedureand

UnderstandingShift the focus to ensure

mathematical understanding over

computation skills

Standards for Mathematical PracticeStudents 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.

Video of NYC High School Piloting the CCSS

Watch first 5 minutes on MathSee link in watsonmath.com

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

bySara Delano Moore, Ph.D.

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)

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.

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.

Start with Good ProblemsCharacteristics

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 Mike’s 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?

Make Sense of Problems (part I)

Mathematically proficient students…Explain the meaning of the problem

to themselvesLook for entry points to the solutionAnalyze givens, constraints,

relationships, goals

Mike’s Canoe Trip– Explain the meaning of the problem– Entry points– Givens, constraints, relationships, goals




• What is the vertical intercept of T? What does this point represent in terms of Mike’s 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?

Persevere in Solving Them

Mathematically proficient students….Plan a solution pathwayConsider analogous cases and

alternate formsMonitor progress and change course

if necessary

Persevere in Solving Them

“It's not that I'm so smart, it's just that I stay with problems longer.”

- Albert Einstein

Mike’s Canoe Trip– Possible solution pathways/strategies– Consider analogous cases & alternate forms– Monitor progress and change course if needed







Make Sense of Problems (part II)

Mathematically proficient students…Explain correspondence and search

for trendsCheck their answers using alternate

methodsContinually ask themselves, “Does

this make sense?” Understand the approaches of others

What can teachers do?• Select rich mathematical tasks

– Connected to rigorous mathematics content

Resources for rigorous mathematical tasks can be found onwww.watsonmath,.com

“North Country High School Math Inservice October 19, 2012”

Illustrative MathematicsMARS TasksInside Mathematics3 – Act Math TasksDan MeyerAndrew StadelOthers

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 you’ve done so far?

What else is there to do?

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.

Attend to Precision

– In Vocabulary– In Mathematical Symbols– In Computation– In Measurement– In Communication

How is the teacher ensuring that students are making sense of problem

and attending to precision?

See Video: Discovering Properties of Quadrilaterals

onWatsonmath.com

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

Mike’s Canoe Trip– Vocabulary– Mathematical Symbols– Computation & Measurement– Communication







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.

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

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

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 them…but not too many!

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 it’s 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)

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 remark—or restating it in their own words—and 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 O’Connor (Boston University), Megan Williams Hall (University of Pittsburgh), with Lauren B. Resnick (University of Pittsburgh)

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...?”


Teacher Moves in Group Discussion

Teacher can model what desirable behaviors and habits of mind look like,

”Here’s what good problem solvers do when they're monitoring their own process."

Teachers can focus the group's thinking by recapping or summarizing key points that have been brought up in a discussion.


Teacher Moves That Support Accountability to the Learning Community

• Accountability to the learning community requires that students listen to one another, attending carefully so that they can use and build on one another's ideas. Students and teachers agree and disagree respectfully, challenging a claim, not the person who made it.

• To support this kind of accountability, teachers must establish a classroom environment where everyone can hear each other, and where everyone knows how important it is to hear and be heard.


Teacher Moves That Support Accountability to the Learning Community

• Keeping the channels open: "Did everyone hear that?"

• Keeping everyone together: "Who can repeat...?"

• Linking contributions: "Who wants to add on...?"

• Verifying and clarifying: "So, are you saying...?”

Teacher Moves That Support Accountability to Accurate Knowledge

• Pressing for accuracy: "Where can we find that?"

• Building on prior knowledge: "How does this connect?"

See slides 41 – 80 on

10-19-12 HS CCSSM Powerpoint Part II

Documents

North Country Inservice HS Mathematics