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Making Math MoreMeaningful with Models

(MMMMM)Duane Habecker

Mathematics Instructional CoachPleasanton Unified School District

@dhabecker

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What is this session about?The premise:

Too often we push our students directly to abstract algorithms without first giving students the prerequisite experience with models such as empty number lines, number bonds, arrow method, area model, etc.

The objective:

Teachers will learn how to use a variety of mathematical representations to make math meaningful for their students.

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Standards for Mathematical Practice

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Key Shifts in Mathematics

● Greater focus on fewer topics

● Coherence: Linking topics and thinking across grades

● Rigor: Pursue conceptual understanding, procedural skills and fluency, and application with equal intensity

What is fluency?

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Procedural fluency...

Procedural fluency builds on a foundation of conceptual understanding, strategic reasoning, and

problem solving.

(NGA Center & CCSSO, 2010; NCTM, 2000, 2014)

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The three phases develop conceptual understanding

The three phases ...to learn basic facts...to develop any conceptual understanding

Phase 1 Modeling and/or counting all or counting on to find the answer; using fingers to help keep track of their counts to solve 5 + 7 = ?

Concrete

Phase 2 Deriving answers using reasoning strategies based on known facts, such as solving 5 + 7 by using doubles or doubles-plus-1

Pictorial

Phase 3 Mastery or efficient production of answers. For example, a child might say, “5 + 7 is 12 because I just know it.”

Abstract

Phase 3 should develop out of meaningful experiences with phase 2.

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Two camps...

Almost all who have ever fully understood arithmetic have been obliged to learn it over again in their own way.

Warren Colburn (1821)

In mathematics, you don’t understand things, you just get used to them.

John Von Neumann (1979)

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Models and representations

The purpose of models and representations is to make sense of something.

Sometimes, it is the algorithm…

...sometimes it is a general mathematical truth.

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Has Duane done a Number Talk yet?

If not, he should do it now.

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Number Bonds K/1 2/3 4/5

Addition 7 + 8 44 + 35 5/9 + 7/9

Subtraction 12 - 5 43 - 25 1 ⅖ - ⅘

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Number Line

K/1 2/3 4/5

Addition 7 + 8 44 + 35 5/9 + 7/9

Subtraction 12 - 5 43 - 25 1 ⅖ - ⅘

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Number Disks

K/1 2/3 4/5

Addition 7 + 8 44 + 35 245 + 466

Subtraction 12 - 5 43 - 25 213 - 158

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Place Value Chart K/1 2/3 4/5

Addition 7 + 8 44 + 35 245 + 466

Subtraction 12 - 5 43 - 25 213 - 158

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Unique models to make math concepts meaningful How to Implement A Story of Units

Array and Area Models Base-Ten Blocks Bundles Money

Number Bond Number Disks Number Line Number Path

Number Towers Place Value Chart Rekenrek Tape Diagram

Ten-Frame Arrow method DotsMore dots Chips

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An amazing resource for ExplainEverything usersLake Elsinore Unified School District

http://www.leusd.k12.ca.us/apps/pages/index.jsp?uREC_ID=324574&type=d&pREC_ID=742299