Principal's Workshop: How Does the Common Core Change

What We Look For in the Math Classroom?

Panama City, FloridaJanuary 22 & 23, 2013

Presenter: Elaine Watson, Ed.D.

Hunt Institute Video: The Importance of Mathematical Practice

http://vimeo.com/album/1702025/video/29568008

1. Make Sense of Problems and Persevere in Solving

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

Albert Einstein

1. Make Sense of Problems and Persevere in Solving

5th Grade Perseverance

1. Make Sense of Problems and Persevere in Solving

Mathematically proficient students:

• Explain to self the meaning of a problem and look for entry points to a solution

• Analyze givens, constraints, relationships and goals• Make conjectures about the form and meaning of

the solution

1. Make Sense of Problems and Persevere in Solving

Mathematically proficient students:

• Plan a solution pathway rather than simply jump into a solution attempt

• Consider analogous problems• Try special cases and simpler forms of original

problem

1. Make Sense of Problems and Persevere in Solving

Mathematically proficient students:

• Monitor and evaluate their progress and change course if necessary…

• “Does this approach make sense?”

1. Make Sense of Problems and Persevere in Solving

Mathematically proficient students:

Persevere in Solving by:

• Transforming algebraic expressions

• Changing the viewing window on a graphing calculator

• Moving between the multiple representations of:

Equations, verbal descriptions, tables, graphs, diagrams

1. Make Sense of Problems and Persevere in Solving

Mathematically proficient students:

• Check their answers• “Does this answer make sense?”

• Does it include correct labels?• Are the magnitudes of the numbers in the solution in the

general ballpark to make sense in the real world?

1. Make Sense of Problems and Persevere in Solving

Mathematically proficient students:

• Check their answers• Verify solution using a different method• Compare approach with others:

• How does their approach compare with mine?• Similarities• Differences

2. Reason Abstractly and QuantitativelyMathematically proficient students:• Make sense of quantities and their relationships in a problem situation• Bring two complementary abilities to bear on problems involving quantitative

relationships: The ability to… decontextualize

to abstract a given situation, represent it symbolically, manipulate the symbols as if they have a life of their own

contextualizeto pause as needed during the symbolic manipulation in order to look

back at the referent values in the problem

2. Reason Abstractly and QuantitativelyMathematically proficient students:

Reason Quantitatively, which entails habits of:• Creating a coherent representation of the problem at hand

considering the units involved• Attending to the meaning of quantities, not just how to

compute them• Knowing and flexibly using different properties of operations

and objects

Watch the video and note where you see evidence of Middle School Classifying Equations1. Make Sense of Problems and Persevere in Solving2. Reason Abstractly and Quantitatively

3.Construct viable arguments and critique the reasoning of others

Mathematically proficient students:

Understand and use… stated assumptions, definitions, and previously established results…

when constructing arguments

3.Construct viable arguments and critique the reasoning of others

• In order for students to be practicing this standard, they need to be talking to each other, so teachers need to plan lessons that include a lot of large group and small group discussions.

• A classroom culture must be cultivated in which it is as safe to disagree as it is to agree.

3.Construct viable arguments and critique the reasoning of others

• Here are some sentence structures from the first video:• I agree that ______________ because __________• I disagree because __________________________• How can we be sure?• What do you think?• Are you convinced?• Do we all agree?

3.Construct viable arguments and critique the reasoning of others

What SMPs do you observe the students practicing?

Here's the Problem from the video:Write several different types of equations for 2.4.

Draw some different types of pictures to represent 2.4.

Is 2.4 the same thing as the quotient 2 remainder 4?

Why or why not?http://youtu.be/EA3YkawKEWc

4. Model with Mathematics

Modeling is both a K - 12 Practice Standard

and a 9 – 12 Content Standard.

4. Model with MathematicsMathematically proficient students:

Use powerful tools for modeling:Diagrams or graphs

SpreadsheetsAlgebraic Equations

4. Model with MathematicsMathematically proficient students:

Models we devise depend upon a number of factors:• How precise do we need to be?• What aspects do we most need to undertand,

control, or optimize?• What resources of time and tools do we have?

4. Model with MathematicsMathematically proficient students:

Models we devise are also constrained by:• Limitations of our mathematical, statistical, and

technical skills• Limitations of our ability to recognize significant

variables and relationships among them

Modeling Cycle

The word “modeling” in this context is used as a verb that describes the process of transforming a real situation into an abstract mathematical model.

Modeling Cycle

Problem Formulate

Compute Interpret

Validate Report

Modeling Cycle

Problem• Identify variables in the situation

• Select those that represent essential features

Modeling Cycle

FormulateSelect or create a geometrical, tabular, algebraic, or

statistical representation that describes the relationships between the variables

Modeling Cycle

ComputeAnalyze and perform operations on these relationships to

draw conclusions

Modeling Cycle

InterpretInterpret the result of the mathematics in terms of the

original situation

Modeling Cycle

ValidateValidate the conclusions by comparing them with the

situation…

Modeling Cycle

Validate

Re - Formulate

Report on conclusions and

reasoning behind them

Modeling Cycle

Problem Formulate

Compute Interpret

Validate Report

6. Attend to precisionMathematically proficient students:

Try to communicate precisely to others:• Use clear definitions• State the meaning of symbols they use• Use the equal sign consistently and appropriately• Specify units of measure• Label axes

6. Attend to precisionMathematically proficient students:

Try to communicate precisely to others• Calculate accurately and efficiently• Express numerical answers with a degree of

precision appropriate for the problem context• Give carefully formulated explanations to each other• Can examine claims and make explicit use of

definitions

6. Attend to precision

Students Practicing and Discussing Precision

Use the Standards for Mathematical Practice Lesson Alignment Template.

What SMPs do you see?

http://ummedia04.rs.itd.umich.edu/~dams/umgeneral/seannumbers-ofala-xy_subtitled_59110_QuickTimeLarge.mov

7. Look for and make use of structureMathematically proficient students:

• Look closely to discern a pattern or structureIn x2 + 9x + 14, can see the 14 as 2 x 7 and the 9 as 2 + 7

• Can see complicated algebraic expressions as being composed of several objects:

5 – 3 (x – y)2 is seen as 5 minus a positive number times a square, so its value can’t be more than 5 for any real numbers x and y

8. Look for and express regularity in repeated reasoning.

Mathematically proficient students:

• Notice if calculations are repeated• Look for both general methods and for shortcuts• Maintain oversight of the process while attending to

the details.

What Practice Standards do you see?

Sean: Is 6 Even and Odd?

Do All 8 Practice Standards Need to be Used in Every Lesson?

There are some rich problems that elicit all 8 of the Practice Standards. However, these types of problems can’t be done on a daily basis. Instructional time still needs to be balanced between building the students’ technical skills and

No…but the teacher should plan so that over the span of a few days, the students are given learning opportunities to of the practicing standards

A Balanced Approach math facts how to approach

and a novel situation procedures mathematically

Math Facts and Procedures

Memorizing Math Facts and Naked Number Exercises are Important!Practice Standards that apply:#2 Reason Quantitatively#6 Attend to Precision#7 Look for and Use Structure#8 Use Repeated Reasoning

Does Every Worthwhile Problem Have to Model a Real World Situation?

What SMPs Do You Observe Maya Practicing? What errors do you

notice? What would you do to have Maya notice the errors?

Maya Representing 52

Let’s Practice Some ModelingStudents can: • start with a model and interpret what it means in

real world terms OR• start with a real world problem and create a

mathematical model in order to solve it.

Possible or Not?

Here is an example of a task where students look at mathematical models (graphs of functions) and determine whether they make sense in a real world situation.

Possible or Not?

Questions:

Mr. Hedman is going to show you several graphs. For each graph, please answer the following:

A. Is this graph possible or not possible?

B. If it is impossible, is there a way to modify it to make it possible?

C. All graphs can tell a story, create a story for each graph.

One

A. Possible or not?

B. How would you modify it?

C. Create a story.

Two

A. Possible or not?

B. How would you modify it?

C. Create a story.

Three

A. Possible or not?

B. How would you modify it?

C. Create a story.

Four

A. Possible or not?

B. How would you modify it?

C. Create a story.

Five

A. Possible or not?

B. How would you modify it?

C. Create a story.

Six

A. Possible or not?

B. How would you modify it?

C. Create a story.

Seven

A. Possible or not?

B. How would you modify it?

C. Create a story.

Eight

A. Possible or not?

B. How would you modify it?

C. Create a story.

Nine

A. Possible or not?

B. How would you modify it?

C. Create a story.

Ten

A. Possible or not?

B. How would you modify it?

C. Create a story.

All 10 Graphs

What do all of the possible graphs have in common?

And now...

For some brief notes on functions!!!!

Lesson borrowed and modified from Shodor.Musical Notes borrowed from Abstract Art Pictures Collection.

Pyramid of Pennies

Here is an example of a task where students look at a real world problem, create a question, and create a mathematical model that will solve the problem.

Dan Meyer’s 3-Act Process

Act IShow an image or short video of a real world situation in which a question can be generated that can be solved by creating a mathematical model.

Pyramid of Pennies 3-Act

http://mrmeyer.com/threeacts/pyramidofpennies/

Dan Meyer’s 3-Act Process

Act I (continued)1. How many pennies are there?

2. Guess as close as you can. 3. Give an answer you know is too high.4. Give an answer you know is too low.

Dan Meyer’s 3-Act Process

Act 2Students determine the information

they need to solve the problem.The teacher gives only the information

students ask for.

Dan Meyer’s 3-Act Process

What information do you need to solve this problem?

Dan Meyer’s 3-Act Process

Act 2 continuedStudents collaborate with each other to create a mathematical model and solve

the problem. Students may need find text or online

resources such as formulas.

Dan Meyer’s 3-Act Process

Go to it!

Dan Meyer’s 3-Act Process

Act 3

The answer is revealed.

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

Procedure and UnderstandingThey shift 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.

Think back to the Pyramid of Pennies. At what point during the problem did you do the following?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.

See Inside Mathematics Videos for good examples of the Practice Standards in

action.

http://www.insidemathematics.org/index.php/mathematical-practice-

standards

Resources for Rich Mathematical Taskshttp://illustrativemathematics.org/

The “Go-To” site for looking at the Content Standards and finding rich tasks, called “Illustrations” that can be used to

build student understanding of a particular Content Standard.

Resources for Rich Mathematical Taskshttp://insidemathematics.org/index.php/home

is a website with a plethora of resources to help teachers transition to teaching in a way that reflects the Standards

for Mathematical Practice.It’s worth taking the 6:19 minutes to watch the

Video Overview of the Video Tours to familiarize yourself to all of the resources. There are more video tours that can be

accessed by clicking on a link below the overview video.

Resources for Rich Mathematical Taskshttp://map.mathshell.org/materials/stds.php

There are several names that are associated with the website: MARS, MAPS, The Shell Center…however the tasks

are usually referred to as The MARS Tasks. The link above will show tasks aligned with the Practice Standards

They have been developed through a partnership with UC Berkeley and the University of Nottingham

Resources for Rich Mathematical Taskshttp://commoncoretools.me/author/wgmccallum/

Tools for the Common Core is the website of Bill McCallum, one of the three principle writers of the CCSSM.

Highlights of this site are the links (under Tools) to the Illustrative Mathematics Project, the Progressions Documents, and the Clickable Map of the CCSSM.

Resources for Rich Mathematical TasksMy blog WatsonMath has a lot of resources listed in the

right hand columnThis powerpoint (with the links removed, but the URLs for the links included) can be accessed on watsonmath.com