Creating a Concept-Based Unit for the 9-12 Mathematics Classroom

October 29, 2009 NCCTM Mathematics Conference

Presented By

Carmella Fair NC Department of Public Instruction Secondary Mathematics Consultant

Agenda

•  Understand the idea of a concept •  Identify concepts •  Know the components of a concept-based

unit

Why Concept-Based Instruction?

•  Increases student understanding

•  Increases student thinking

•  Creates the motivation for learning

Why is the use of concepts important?

•  Changes in society impact education –  Increased use of technology – Change in job demands –  Increase in global interdependence – Change in social norms and structures

Two Dimensional Curriculum Model-

Processes & Skills

Factual Content

(Topic-based)

(Concept-based) Three Dimensional Curriculum Model-

Concepts & Principles

KNOW DO

From H. Lynn Erickson

Attributes of

2-Dimensional Model

Attributes of

3-Dimensional Model

Coverage centered Idea centered

Intellectually shallow Intellectual depth

Inability to transfer factual knowledge

Concepts & generalizations transfer

Students not prepared for 21st Century

Students prepared for a changing world

A Shift in Thinking Making a case for concept-based curriculum & instruction

From H. Lynn Erickson

•  Timeless •  Universal •  Abstract •  Represented by 1 or 2 words •  Examples share common attributes

A concept is an organizing idea

Examples of Mathematics Concepts:

Patterns & Sequences – Linear Functions – Scatter Plots – Graphs –Order – System

From H. Lynn Erickson

Your Conceptual Mind

Identify 2 – 4 mathematical concepts from a course or courses that you teach.

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Mathematics Concepts Rational Numbers Positive Integers Common Factors Common Multiples Proper Factor Factor Pair Square Numbers Prime Composite Expressions

Grade  Three   Grade  Six   Algebra  I  

Whole Numbers Place Value Number Representations Patterns Number Grids Money Symbols Decimal Point Related Addition/ Subtraction Facts Fact Families Mathematical Models Standard Units Non-Standard Units Estimation Converting Units Units of Measure

Independent/Dependent Quantities Functional Relationships Domain/Range Patterns and Sequences Linear Functions Variables and Symbols Algebraic Expressions Order of Operations Distributive Property Coordinate Plane Graphs Linear Parent Functions Scatter Plots

From  H.  Lynn  Erickson  

•  Timeless •  Universal •  Abstract •  Represented by 1 or 2 words •  Examples share common attributes

A concept is an organizing idea

Examples of Mathematics Concepts:

Patterns & Sequences – Linear Functions – Scatter Plots – Graphs –Order – System

From H. Lynn Erickson

Science Social Studies Mathematics Music Order Conflict Order Rhythm Organism Patterns Ratio Melody Population Populations Proportion Harmony System System System Tone Change Change/Continuity Pattern Pattern Evolution Culture Probability Form Cycle Evolution Function Tempo Interaction Civilization Slope Timbre Energy/Matter Interdependence Graph Pitch

From H. Lynn Erickson

5 Things That Concept-Based Curriculum & Instruction Does

Unites “big ideas” for deeper understanding

Aligns curriculum, instruction and assessment

Emphasizes & identifies what students should Know, Understand & Do

Organizes the concepts/content into instructional units/lessons that contextualize the standards

Identifies concepts in curriculum rather than just facts

Concept-based Instruction Teaching using concepts as a tool to help students see patterns and connections between facts and related ideas in order to reach a deeper understanding of the content.

What is written

What is taught

What is tested

Graphic Organizer for Graphs

Graphs

In small groups use the graphic organizer to identify characteristics that make graphs - the concept “graphs.”

Macro and Micro Concepts

Macro Concepts provide BREADTH of understanding

(interdisciplinary)

Micro Concepts provide DEPTH of understanding

(discipline specific)

Macro and Micro Concepts Macro Concepts provide BREADTH of understanding

(interdisciplinary)

Micro Concepts provide DEPTH of understanding

(discipline specific)

Example:

System

System of Equations (mathematics)

Governmental Systems (social studies)

Ecosystem (science)

Identifying Concepts

A. Micro B. Macro

Order

Identifying Concepts

A. Micro B. Macro

Rational Numbers

Identifying Concepts

A. Micro B. Macro

System

Identifying Concepts

A. Micro B. Macro

Common Factor

Identifying Concepts

Step 1: Identify the focus of a Concept-based unit of Instruction.

Step 2: What standards will be taught/assessed?

Step 3: What concepts (both explicit and implied) will be addressed in this unit?

Identifying Concepts

Unit Title/Focus:

NCSCOS Standard(s): Math A

Algebra I/IM I Math BC

Geometry/Algebra II IM II/IM III

Concepts: Math A

Algebra I/IM I Math BC

Geometry/Algebra II IM II/IM III

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The  Structure  of  Knowledge  

From  H.  Lynn  Erickson  

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The Structure of Knowledge

slope line

Measurement of distance and speed

Velocity can be represented mathematically by the slope of a line.

The slope of a graph at a particular point indicates the instantaneous rate of change.

graph

From  H.  Lynn  Erickson  

Concepts and RBT Knowledge is: – Factual – Conceptual – Procedural – Metacognitive

Mathematics is concept-based, but we need to ensure that students attain conceptual understanding.

A Conceptual Mind: •  Creates connections to prior experience •  Works with factual knowledge to develop the

intellect •  Creates deeper understanding at factual and

conceptual levels •  Recognizes the transferability of knowledge •  Creates the motivation for learning

References

Erickson, H. L. (2008). Stirring the head, heart, and soul: Redefining curriculum, instruction, and concept-based learning (3rd ed.). Thousand Oaks, CA: Corwin Press.

Erickson, H. L. (2007). Concept-based curriculum and instruction for the thinking classroom. Thousand Oaks, CA: Corwin Press.

Contact Information

Carmella Fair NC Department of Public Instruction Secondary Mathematics Consultant 919-807-3840 cfair@dpi.state.nc.us