HABITS OF MIND - MATHEMATICAL

PRACTICES OF THE COMMON CORE

Statewide Instructional Technology Project

How many of your feel the same?

http://www.tubechop.com/watch/303621

Introduction

An introductory webinar was offered March 6th. Please watch the recording for the explanation introducing Common Core and the changes to the AZ 2010 Math Standards.

To view the official ADE documents 2010 Arizona Mathematics Standards

Overview of the 2010 Mathematical Standards PDF

Standards for Mathematical Practices PDF

Mathematics Introduction (Coming Soon)

Mathematics Glossary PDF Summary of Updates to Explanations

and Examples PDF

Gauge the Audience

What grade level do you teach? A = K-5 B = 6-8 C = 9-12

http://www.tubechop.com/watch/290412

How does your classroom compare?

The Mathematical Practices10

The Mathematical Practices11

1. Make sense of problems and persevere in solving them

6. Attend to precision

The Habits of Mind of a

Productive Mathematical Thinker

MP1. Make sense of problems and persevere in solving them

In Kindergarten, students begin to build the understanding that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Younger students may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, “Does this make sense?” or they may try another strategy.

In first grade, students realize that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Younger students may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, ―Does this make sense? They are willing to try other approaches.

MP1. Make sense of problems and persevere in solving them

In first grade, students realize that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Younger students may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, ―Does this make sense? They are willing to try other approaches.

In second grade, students realize that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. They may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, ―Does this make sense? They make conjectures about the solution and plan out a problem-solving approach.

MP1. Make sense of problems and persevere in solving them

In second grade, students realize that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. They may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, ―Does this make sense? They make conjectures about the solution and plan out a problem-solving approach.

In third grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Third graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, ―Does this make sense? They listen to the strategies of others and will try different approaches. They often will use another method to check their answers.

MP1. Make sense of problems and persevere in solving them

In third grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Third graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, ―Does this make sense? They listen to the strategies of others and will try different approaches. They often will use another method to check their answers.

In 4th grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Fourth graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, “Does this make sense?” They listen to the strategies of others and will try different approaches. They often will use another method to check their answers.

MP1. Make sense of problems and persevere in solving them

In 4th grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Fourth graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, “Does this make sense?” They listen to the strategies of others and will try different approaches. They often will use another method to check their answers.

In 5th grade, students solve problems by applying their understanding of operations with whole numbers, decimals, and fractions including mixed numbers. They solve problems related to volume and measurement conversions. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, ―What is the most efficient way to solve the problem?, ―Does this make sense?, and ―Can I solve the problem in a different way?

Make sense of problems and persevere in solving them

In 5th grade, students solve problems by applying their understanding of operations with whole numbers, decimals, and fractions including mixed numbers. They solve problems related to volume and measurement conversions. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, ―What is the most efficient way to solve the problem?, ―Does this make sense?, and ―Can I solve the problem in a different way?

In grade 6 grade, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”.

MP1. Make sense of problems and persevere in solving them

In grade 6 grade, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”

In grade 7, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”.

MP1. Make sense of problems and persevere in solving them

In grade 7, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”.

In grade 8, students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”

MP1. Make sense of problems and persevere in solving them

High school students start to examine problems by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals.

They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution.

They monitor and evaluate their progress and change course if necessary.

Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. By high school, students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.

They check their answers to problems using different methods and continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Good problems:

Mathematics questions or tasks that are challenging enough so it becomes useful to take in ideas other than just one’s own to engage thinking.

Productive Thinking vs Reproductive Thinking Connect to student experience or interest Incorporates rich mathematics Entry points solutions pathways are not obvious

MP1. Make sense of problems and persevere in solving them

Mathematically proficient students make sense of problems:

Mathematically proficient students persevere in problem solving:

Make sense of problems Plan a solution pathway

Explain the meaning of the problem to themselves

Consider similar cases and alternate form

Look for entry point Monitor progress and change course if necessary

Analyze givens, constraints, relationships, goals

Explain correspondence and search for trendsCheck their answers using alternate methodsContinually ask themselves, “Does this make sense?” Listen to and work to understand the approaches of others

MP 6. Attend to precision

Mathematically proficient students:Communicate precisely to others. Use clear definitions in discussion with others and in their own reasoning.State the meaning of the symbols they chooseSpecify units of measure, and labeling axes to clarify the correspondence

with quantities in a problem. Calculate accurately and efficiently, express numerical answers with a

degree of precision appropriate for the problem context. Give carefully formulated explanations to each other.In high school, they have learned to examine claims and make explicit

use of definitions.

Technology Resources

To solve this problem you might explore (Connecting Cubes) http://www.eduplace.com/kids/mw/manip/mn_2.html

(Pattern Blocks) http://gingerbooth.com/flash/patblocks/patblocks.php

(Geoboards) http://nlvm.usu.edu/en/nav/frames_asid_129_g_2_t_3.html?open=activities

(Graphs) http://nces.ed.gov/nceskids/createagraph/default.aspx

Technology Resources

A Short Introduction to Using the Connecting Cubes eManipulatives(Click to play)

Technology Resources

A Short Introduction to Using the Virtual Pattern Blocks(Click to play)

Technology Resources

A Short Introduction to Using the Virtual Geoboard(Click to play)

Discuss how you solved the problem

After solving the problem, complete the form linked (https://docs.google.com/spreadsheet/viewform?formkey=dHZQaXZDUmxOdzFENV93UkpqOXBMNmc6MQ#gid=0

) in the chat area to answer the following questions:

1.What answer did you get?2.Which technology tool did you use?

3.Why did you choose this tool?4.How did this tool help you solve the problem?

Solutions to the Problem

Click https://docs.google.com/spreadsheet/ccc?key=0Aoxuu3aQOMKNdHZQaXZDUmxOdzFENV93UkpqOXBMNmc to access the results of the form.

Problem Solving w/No Technology

Problem Solving with Technology

Problem Solving with Technology

Problem Solving with Technology

Problem Solving with Technology

.

Student Example: Problem Solving

Number of Tables Number of Seats

1 5

2 8

3 11

4 14

5 17

… …

20 62

I found there would be 17 seats at 5 tables. I noticed that each time I added a table, the number of seatsincreased by three. That is because we are adding five new places but losing two on the sides of thetables that connect.

To answer question 2, I counted out 15 more trapezoids to make a total of 20 tables. Then I skip countedby threes for each new table, starting with 17, until I came to 62 seats for the 20th table.

Student 1: List created after student modeled with pattern blocks.

Student Example: Problem SolvingStudent 2: (Draw a diagram, then direct calculation)I used pattern blocks to help me see the pattern. After adding several

tables I discovered that each table added three seats to the total. At the ends there were always two more seats, no matter how long the row was. Here are my results: (organized list)

To answer question 2, I figured out how many more tables I would need: 20 - 5 = 15 more tables

Each of those 15 new tables would add three seats to the row: 15 * 3 = 45 more seats

I added the seats from 5 tables and the new seats: 17 + 45 = 62 total seats at 20 tables

Student Example: Problem Solving

Student 3: (Direct Calculation based on seats lost)

I multiplied the number of tables by the number of people who can sit at one table:

5 * 5 = 25

For each of the places where two tables connect, we lose 2 seats. There are 4 of those places, one less than the total number of tables. 4 * 2 = 8

I subtracted the number of seats lost from the total places:

25 – 8 = 17 seats at 5 tables

I did the same thing for 20 tables:

20 * 5 = 100 total seats

19 * 2 = 38 seats lost at the connections

100 – 38 = 62 seats at 20 tables

Use words or numbers and symbols to write a rule for calculating the number of volunteers that can sit at any given number of tables.

How many tables would it take, arranged in a straight line, to seat 85 volunteers?

Extend the Problem

Extend the Problem

Discuss your attention to precision Communicate terms and thinking

process Link to Spreadsheet from form.

Mathematical Practices

Expertise that we each seek to develop in our students What does it mean to do mathematics? What does it mean to understand mathematics?

As teachers, our goal is to provide regular and consistent opportunities to develop and build these habits of mathematical thinking.

1st Steps to Implementation of Mathematical Practices

How to transition Begin with the Mathematical Practices of CCS Look closely at the Critical Areas provided for

each grade

What shall I do first Focus on the Mathematical Practices

How do your students model the Mathematical Practices?

In what ways do your classroom strategies foster development of the Mathematical Practices?

Implement the Critical Ideas Look at the ADE website for standards, crossw

alk, and summary of changes.

Resources for Further Exploration The Illustrative Mathematics Project

http://illustrativemathematics.org/ Math Common Core Coalition

http://www.nctm.org/standards/mathcommoncore/ Achieve the Core http://www.achievethecore.org/ National Council Teacher of Mathematics

http://www.nctm.org/standards/content.aspx?id=23273 Guiding Principles for Mathematics Curriculum and Assessment

Mathematics Problem Solving http://jwilson.coe.uga.edu/emt725/PSsyn/Pssyn.html

Mathematics Through Problem Solving http://www.mathgoodies.com/articles/problem_solving.html

Resources for Further Exploration Inside Mathematics

http://www.insidemathematics.org/index.php/standard-1

Curriculum Exemplars from EngageNY http://engageny.org/resource/curriculum-exemplars/

Indiana Dept of Ed Implementing the Standards for Mathematical Practice

Indiana Dept of Ed - Implementing the Standards for Mathematical Practice http://media.doe.in.gov/commoncore/2011-05-10-StandforMath.html

Tools for the Common Core Standards http://commoncoretools.me/

Resources for Further Exploration New Jersey Center for Teaching & Learning

Progressive: Mathematics Initiative http://njctl.org/programs/ Free digital course content for over twenty courses, these initiatives span K-12 mathematics and high school science.

Rich problem source (middle/HS) http://psc.stanford.edu/psc-materials/problem-bank.html

Kenken http://www.kenken.com/index.html Math Forum http://mathforum.org/kenken/ Online Pattern Blocks

http://gingerbooth.com/flash/patblocks/patblocks.php Wiki on Standards of Practice

http://enhancingmypractice.wikispaces.com/Standards+of+Math+Practice

To view the official ADE documents 2010 Arizona Mathematics Standards

Overview of the 2010 Mathematical Standards PDF

Standards for Mathematical Practices PDF

Mathematics Introduction (Coming Soon)

Mathematics Glossary PDF Summary of Updates to Explanations

and Examples PDF

More Questions

Contact ADE Mary Knuck Mary.Knuck@azed.gov Suzi Mast suzi.mast@azed.gov