Montville Township Public SchoolsMiddle and High School

Introduction toCommon Core State Standards for

Mathematics

Monday, October 8, 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?

Goals for this WorkshopYou will leave with a deeper understanding of:• The Content Standards & Practice Standards• The types of tasks that build students’ ability to “practice” the Practice Standards• How to navigate rich online CCSSM resources• How to practice formative instruction to reach

the different learners in your class

What are the standards, anyway?The standards are the end product of what students need to understand and be able to do with the content. They do not attempt to tell teachers how to teach the content. Teaching is an art. There is no one method that fits all teachers or all students.

CCSSM Equally Focuses on…

Standards for Mathematical

Practice

Standards for Mathematical

Content

Same for All Grade Levels

Specific to Grade Level

CCSSM Video

The Mathematical Standards: How They Were Developed and Who Was Involved

Modelingis both a

K - 12 Practice Standard and a

9 – 12 Content Standard.

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

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.

CCSSM VideoHigh School Math Courses

Students 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.

Dan Meyer’s 3-Act Math Task

Pyramid of Pennies

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.

Dan Meyer’s 3-Act Process

Act 3

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.

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.

Content Standard ActivityWork with a partner and on your own laptop, go to

http://illustrativemathematics.org/• Go to at least one content domain and read

through some standards that have illustrations.• Check out the illustrations. • How does the illustration assess the understanding

of the standard?• How could you use the illustrations in your class?

Formative Instruction

How do we know if our students know what we want them to know?

Formative InstructionFormative instructional practices are not new—this work

dates back to Benjamin Bloom and mastery learning. Teachers "do" formative instruction every day in their

classrooms, but for many teachers it is not ongoing and intentional. The latest research reveals that formative

instructional practices are effective in closing the achievement gap. These practices are proven to raise the achievement of all students—with the greatest gains for

low-achieving students.

Formative InstructionAccording to brain research, we know that growth is not fixed; it is a mindset. We have an innate desire to learn. This intrinsic motivation to learn is supported when the student has a sense of control and choice,

gets frequent and specific feedback about where he/she is, encounters learning that is challenging but not threatening, is able to self-assess accurately and

encounters real-life learning tasks.

Formative InstructionThese “supports” are formative instructional practices. Formative instructional practices

provide students with opportunities for penalty-free learning, so when it is time to

measure (summative assessment), students feel in control of their success.

Examples of Formative AssessmentAnecdotal Notes: These are short notes written during a lesson as students work in groups or individually, or after the lesson is complete. The teacher should reflect on a specific aspect of the learning (sorts geometric shapes

correctly) and make notes on the student's progress toward mastery of that learning target. The teacher can create a

form to organize these notes so that they can easily be used for adjusting instruction based on student needs.

Examples of Formative AssessmentExit Slips are written responses to questions the teacher poses at the end of a lesson or a class to assess student understanding of

key concepts. They should take no more than 5 minutes to complete and are taken up as students leave the classroom. The

teacher can quickly determine which students have it, which ones need a little help, and which ones are going to require much more instruction on the concept. By assessing the responses on

the Exit Slips the teacher can better adjust the instruction in order to accommodate students' needs for the next class.

Examples of Formative AssessmentAsking students questions about their reasoning and

promoting rich classroom discourse between students creates an opportunity for student to make sense of

problems, construct viable arguments, and critique the reasoning of others. It also provides teachers with

significant insight into the degree and depth of student understanding. Questions should go beyond the typical factual questions requiring recall of facts or numbers.

Small Group ActivityDiscuss ways that you can use at least one of these strategies in

a lesson during the next week. Commit to at least one:

• Keeping Anecdotal Notes• Using Exit Slips and following up on what the

data tell you • Rich questioning and encouraging student to

student discourse.

Goals for this WorkshopYou will leave with a deeper understanding of:• The Content Standards & Practice Standards• The types of tasks that build students’ ability to “practice” the Practice Standards• How to navigate rich online CCSSM resources• How to practice formative instruction to reach

the different learners in your class

Resourceswww.watsonmath.com

“October 8, 2012 Montville Township Middle and High School”on watsonmath.com.