BUILDING CONCEPTUAL KNOWLEDGE IN MATHEMATICS: GRADES 3 - 5Fluency Strategies December 7, 2012Presenters: Andrea Tamarazio & Steven GraserErie 1 BOCES

MAD MINUTE! READY, SET, GO

When asked to begin, take one minute to complete as many questions as you can

ANTICIPATORY GUIDE Read each statement

Circle True / False Write an sentence / comment about each

statement Complete Frayer Model for “fluency”

SPRINT!! READY, SET, GO!

When asked to begin, take one minute to complete as many questions as you can

Sprint Video

SPRINT VS. MAD MINUTE Take a minute to complete the graphic

organizer Likes and Dislikes of each activity

Share Out

MATHEMATICAL SHIFTS

MATHEMATICAL PRACTICES Make sense of problems and persevere in solving

them Reason abstractly and quantitatively Construct viable arguments and critique the

reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated

reasoning

FLUENCY PRACTICE (COMMON CORE INC.)Fluency in designed to promote automaticity by engaging students in practice in ways that get their adrenaline flowing. Automaticity is critical so students avoid using up to many of their attention resources with lower level skills when they are addressing lower level problems. Computational foundations enables deep understanding.

SAMPLE TASK What fluencies do students need to solve this

problem?

See Handout for problem

COMMON CORE STANDARDS Professional Development Needs

Curriculum Assessments Instruction (Teacher Capacity)

What Common Core aligned PD has your district provided in regards to the items listed above?

MATH FLUENCYExperts agree that the ability to recall basic

math facts fluently is necessary for students to attain higher-order math skills.

*Drop off in student performance in the middle and high school years can be attributed to math instruction at the elementary level. Many students memorize facts and can pass elementary assessments, however the lack of computational fluency becomes apparent in the secondary grades.

WARM UP: MENTAL MATH ACTIVITY 7 Questions

Solve mentally Explain the process used to solve each problem

JIGSAW ACTIVITY In small groups, read assigned articles Complete Graphic Organizer for your article

MENTAL MATH Mental Math is one component of Computational

Fluency Understanding Basic Number Combinations

Number Bonds Dot Plates

Understanding Base Ten Number System 10 times more, 10 times less In the number 220, the 2 in the hundreds place is 10 times

greater than the 2 in the tens place Recognize and Use related problems

Fact Families Inverse Functions; solve for missing addend, subtrahend, and

minuend. Understand and Use Properties

Commutative and Associative Making Connections between operations

Inverse Functions

MATH MODELS Place Value & Standard Algorithm

Bundles Place Value Chart Base Ten Blocks Money Number Disks with Place Value Chart

Fractions Number Line Area Model

Multiplication Array Area Model Rekenrek (Number Rack)

Word Problems Bar Diagram

FLUENCY VS. MEMORIZATION Memorization

Single procedure and / or recall

Fluency Understanding of operations and their

relationship to each other (inverse function; fact families)

Number Relationships (compatible numbers – fives and tens)

Base Ten System For Example: recognizing the relationship between 5

+ 9 and 50 + 90

COMPUTATIONAL FLUENCYComputational Fluency includes three ideas: efficiency,

accuracy, and flexibility Efficiency implies that the student does not get bogged down

in too many steps or lose track of the logic of the strategy. An efficient strategy is one that the student can carry out easily, keeping track of subproblems and making use of intermediate results to solve the problem.

Accuracy depends on several aspects of the problem-solving process, among them careful recording, knowledge of number facts and other important number relationships, and double-checking results.

Flexibility requires the knowledge of more than one approach to solving a particular kind of problem, such as two-digit multiplication. Students need to be flexible in order to choose an appropriate strategy for the problem at hand, and also to use one method to solve a problem and another method to double-check the results.

ASSESSMENT OF COMPUTATIONAL FLUENCY The student draws on basic facts and number

relationships Properties

Student uses the structure of the base ten system 2 + 5 = 7; 20 + 50 = 70; 200 + 500 = 700

Student recognizes related problems Students relationships among related operations Student knows what each number and numeral means in

a problem Student can explain the steps he or she uses to solve

even recall problems Same student explains twice

Student can clearly record and track his or her procedure Student may have multiple approaches to solve a

problem

“DRILL & KILL” ≠ COMPUTATIONAL FLUENCY Memorization of math facts is not the same as

understanding number concepts Regardless of time and energy spent with “drill &

kill”, some students will not master facts Memory

Memory problem interferes with ability to retrieve basic facts quickly

Short Term Retention Cognitive

Understanding relationship between numbers Solving word problems Understanding number system & effective counting strategies

Visual / Spatial Misalignment of numerals in a column Problems with Place Value

COMMON CORE FLUENCY

FACT FLUENCY INSTRUCTIONAL FLOW Number Relationships & Concepts

40 – 50% of Instructional Time Contextualizing math fact problems Creating arrays and drawing problems Decomposing or using known facts Using compensation strategies

Fact Strategy Development 30 - 40% of Instructional Time Doubles, Near Doubles, Combinations to 10

Drill 15 – 20% of Instructional Time Fact Fluency related games and flash cards

Fluency Assessment Less than 3% of Instructional Time Timed Test

CCLS FLUENCY RECOMMENDATION

Distribution of Instructional Minutes

NUMBER RELATIONSHIPS & CONCEPTS Contextualize Math Facts

Reading, Writing, and Symbolizing Math Problems High Number Toss Game

Creating Arrays and Drawing Pictures (word to picture; number to picture; picture to word or number)

Place Value Literacy Decomposing Numbers & Listing their Values Number Line Up Build a Number

FLUENCY ACTIVITIES Adding Mentally

Whole Numbers, Fractions & Decimals 1, 2, 3, sit on 20

This can be modified for multiple grade levels – skip counting, multiplication facts, etc

Take the 1 Out Fluency Counting

This can be modified for multiple grade levels Skip Counting by Fractions White Board Equivalent Fractions

Written or Oral

FACT STRATEGY DEVELOPMENT (CHORAL WHITE BOARD) One-More-Than & Two-More-Than Facts

Facts with Zero

Doubles / Near Doubles (Doubles Plus 1, Doubles Plus 2)

Compensation (Plus 1, Minus 1; Plus 2 Minus 2)

Compatible Numbers (Combinations to 5 or 10)

DRILLS Around the World

Math Fact Race

Math Fact War

BINGO

I Have You Have

ACTIVITY SHARE OUT Take time to create activities to use in your

classroom

What do you have in your classroom that you would like to share? Activities

Strategies

QUESTIONS / COMMENTS If you have any questions or comments,

please feel free to contact us.

Andrea Tamarazioatamarazio@e1b.org

Steven Grasersgraser@e1b.org