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Learner Profile Card

Gender Stripe

Auditory, Visual, Kinesthetic

Modality

Analytical, Creative, Practical

Sternberg

Student’s Interests

Multiple Intelligence Preference

Gardner

Array Inventory

Array Interaction Inventory

- Directions:
- Rank order the responses in rows below on a scale from 1 to 4 with 1 being “least like me” to 4 being “most like me”.
- After you have ranked each row, add down each column.
- The column(s) with the highest score(s) shows your primary Personal Objective(s) in your personality.

Personal Objectives/Personality Components

Teacher and student personalities are a critical element in the classroom dynamic. The Array Model (Knaupp, 1995) identifies four personality components; however, one or two components(s) tend to greatly influence the way a person sees the world and responds to it. A person whose primary Personal Objective of Production is organized, logical and thinking-oriented. A person whose primary Personal Objective is Connection is enthusiastic, spontaneous and action-oriented. A person whose primary Personal Objective is Status Quo is insightful, reflective and observant. Figure 3.1 presents the Array model descriptors and offers specific Cooperative and Reluctant behaviors from each personal objective.

the learners inherit the earth

while the learned find

themselves beautifully equipped

to deal with a world that

no longer exists.”

Eric Hoffer

Know

These are the facts, vocabulary, dates, places, names, and examples you want students to give

you.

The know is massively

forgettable.

“Teaching facts in isolation is like trying to pump water uphill.”

-Carol Tomlinson

Facts 2X3=6,

Vocabulary numerator, slope

KNOW (Facts, Vocabulary, Definitions)

- Definition of numerator and denominator
- The quadratic formula
- The Cartesian coordinate plane
- The multiplication tables

Skills

- Basic skills of any discipline
- Thinking skills
- Skills of planning, independent learning, etc.
- The skill portion encourages the students to “think” like the professionals who use the knowledge and skill daily as a matter of how they do business. This is what it means to “be like” a mathematician, an analyst, or an economist.

Research about teaching suggests that learning by struggling at first with a concept enables students to benefit from an explanation that brings the ideas together (Schwartz & Bransford, 2000).

BE ABLE TO DO (Skills: Basic Skills, Skills of the Discipline, Skills of Independence, Social Skills, Skills of Production)

- Describe these using verbs or phrases:
- Analyze, test for meaning
- Solve a problem to find perimeter
- Generalize your procedure for any situation
- Evaluate work according to specific criteria
- Contribute to the success of a group or team
- Use graphics to represent data appropriately

Certain methods of teaching, particularly those that emphasize memorization as an end in itself tend to produce knowledge that is seldom, if ever, used. Students who learn to solve problems by following formulas, for example, often are unable to use their skills in new situations. (Redish, 1996)

It Begins with Good Instruction

Adding It Up (National Research Council) – Rule-based instructional approaches that do not give students opportunities to create meaning for the rule, or to learn when to use them, can lead to forgetting, unsystematic errors, reliance on visual clues, and poor strategic decisions.

Research about teaching suggests learning may be hindered by

- isolated sets of facts that are not organized and connected or organizing principles without sufficient knowledge to make them meaningful (NRC, 1999)
- Students have become accustomed to receiving an arbitrary sequence of exercises with no overarching rationale.”(Black and Wiliam, 1998))

Major Concepts and

Subconcepts

These are the written statements of truth, the core to the meaning(s) of the lesson(s) or unit. These are what connect the parts of a subject to the student’s life and to other subjects.

It is through the understanding component of instruction that we teach our students to truly grasp the “point” of the lesson or the experience.

Understandings are purposeful. They focus on the key ideas that require students to understand information and make connections while evaluating the relationships that exist within the understandings.

UNDERSTAND (Essential Truths That Give Meaning to the Topic)

Begin with I want students to understand THAT…

- Multiplication can have different meanings in different contexts, including repeated addition, groups and creation of area.
- Fractions always represent a relationship of parts and wholes.
- Addition and subtraction show a final count of the same thing.
- Functions can be represented in many ways (graphs, words, tables, equations) but all representations are of the same function.

Some questions for identifying truly “big ideas”

- Does it have many layers and nuances, not obvious to the naïve or inexperienced person?
- Do you have to dig deep to really understand its meanings and implications even if you have a surface grasp of it?
- Is it (therefore) prone to misunderstanding as well as disagreement?
- Does it yield optimal depth and breadth of insight into the subject?
- Does it reflect the core ideas as judged by experts?

Hints for Writing Essential Understandings

- Essential understandings synthesize ideas to show an important relationship, usually by combining two or more concepts.
- For example:
- People’s perspectives influence their behavior.
- Time, location, and events shape cultural beliefs and practices.
- Tips:
- When writing essential understandings, verbs should be active and in the present tense to ensure that the statement is timeless.
- Don’t use personal nouns- they cause essential understanding to become too specific, and it may become a fact.
- Make certain that an essential understanding reflects a relationship of two or more concepts.
- Write essential understandings a complete sentences.
- Ask the question: What are the bigger ideas that transfer to other situations.

Concepts

Some concepts

- span across several subject areas
- represent significant ideas, phenomena,intellectual process, or persistent problems
- Are timeless
- Can be represented though different examples, with all examples having the same attributes
- And universal

For example, the concepts of patterns, interdependence, symmetry, system and power can be examined in a variety of subjects or even serve as concepts for a unit that integrates several subjects.

Discipline-based Concepts

- Art-color, shape, line, form, texture, negative space
- Literature-perception, heroes and antiheroes, motivation, interactions, voice
- Mathematics-number, ratio, proportion, probability, quantification
- Music-pitch, melody, tempo, harmony, timbre
- Physical Education-movement, rules, play, effort, quality, space, strategy
- Science-classification, evolution, cycle, matter, order
- Social Science- governance, culture, revolution, conflict, and cooperation

Mortimer Adler’s List of the Most Important Concepts in Western Civilization

- Angel
- Animal
- Aristocracy
- Art
- Astronomy
- Beauty
- Being
- Cause
- Chance
- Change
- Citizen
- Constitution
- Courage
- Custom and convention
- Definition
- Democracy
- Desire
- Dialectic
- Duty
- Education
- Element
- Emotion
- Eternity
- Evolution
- Experience
- Family
- Fate
- Form
- God
- Good and Evil
- Government
- Habit
- Happiness
- History
- Honor
- hypothesis

- 37. Idea
- Immortality
- Induction
- Infinity
- Judgment
- Justice
- Labor
- Language
- Law
- Liberty
- Life and death
- Logic
- Love
- Man
- Mathematics
- Matter
- Mechanics
- Medicine
- Memory/Imagination
- Metaphysics
- Mind
- Monarchy
- Nature
- Necessity
- Oligarchy
- One and Many
- Opinion
- Opposition
- Philosophy
- Physics
- Pleasure and Pain
- Poetry
- Principle
- Progress
- Prophecy
- Prudence

- Punishment
- Quality
- Quantity
- Reasoning
- Relation
- Religion
- Revolution
- Rhetoric
- Same/Other
- Science
- Sense
- Sign/Symbol
- Sin
- Slavery
- Soul
- Space
- State
- Temperance
- Theology
- Time
- Truth
- Tyranny
- Universe
- Virtue/Vice
- War & Peace
- Wealth
- Will
- Wisdom
- World

Mathematical Concepts

Number Error / Uncertainty Ratio

Measurement Proportion Behavior

Symmetry Relationships Pattern

Probability Function Truth

Order Problem solving Change

System Quantification Prediction

Representation

- Mathematical Understandings

Our number system maintains order and is rich with patterns.

Mathematicians quantifydata in order to establish real-world probabilities.

All measurement involves error and uncertainty.

Strands of Mathematical ProficiencyAdding It Up, 2001

- Conceptual Understanding
- Procedural Fluency
- Strategic Competence
- Adaptive Reasoning
- Productive Disposition

Strands of Mathematical Proficiency: Adding It Up, 2001

- Conceptual Understanding - Comprehension of mathematical concepts, operations and relations

Strands of Mathematical Proficiency: Adding It Up, 2001

- Conceptual Understanding - “Refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which it is useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention.” P. 118

Strands of Mathematical Proficiency: Adding It Up, 2001

- Procedural Fluency - Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately

Strands of Mathematical Proficiency: Adding It Up, 2001

- “Understanding makes learning skills easier, less susceptible to common errors, and less prone to forgetting. By the same token, a certain level of skill is required to learn many mathematical concepts with understanding.” Page 122

Strands of Mathematical Proficiency: Adding It Up, 2001

- Strategic Competence - Ability to formulate, represent, and solve mathematical problems, especially with multiple approaches.

Strands of Mathematical Proficiency: Adding It Up, 2001

- Adaptive Reasoning - Capacity for logical thought, reflection, explanation, and justification

Strands of Mathematical Proficiency: Adding It Up, 2001

- Productive Disposition - Habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy

Research suggests that learning is enhanced by providing opportunities for

- Struggling
- Choosing and evaluating strategies
- Contrasting cases
- Organizing information
- Making connections (NRC, 1999)

Dividing FractionsDemonstrating the 5 Strands

- What does it mean to divide? What meanings does division have?
- Repeated subtraction
- Partitioning or dividing up into groups
- Measurement (fits into)
- Of these meanings, which one works with dividing fractions?

Dividing FractionsDemonstrating the 5 Strands

- Measurement Model

6 Divided by 2

Can you think of examples where you would need to divide fractions?

Dividing FractionsDemonstrating the 5 Strands

- Dividing fractions with fraction strips

Dividing FractionsDemonstrating the 5 Strands

- Try dividing some fractions with like denominators on your own using the fraction strip model
- Share your findings.
- Do you see a pattern in dividing fractions with like denominators?

Dividing FractionsDemonstrating the 5 Strands

- Do you know a rule that can help speed up the process for dividing fractions without strips?
- Can you think of a way to use the pattern discovered with dividing common denominators to make sense of this rule?

Dividing FractionsDemonstrating the 5 Strands

Write problem with common denominators

Divide the numerators

Dividing FractionsDemonstrating the 5 Strands

- Now relate the pattern to the algorithm of invert and multiply…
- Where does the common denominator come from?

Invert and multiply!

Dividing FractionsDemonstrating the 5 Strands

- How can knowing how to divide fractions help you in your life?
- Think of as many ideas as you can for the benefits of knowing how to divide fractions!

Fraction Activity

- What went well for you?
- What was a challenge for you?
- What did you learn from this activity?

USE OF INSTRUCTIONAL STRATEGIES.

The following findings related to instructional strategies are supported by the existing research:

- Techniques and instructional strategies have nearly as much influence on student learning as student aptitude.
- Lecturing, a common teaching strategy, is an effort to quickly cover the material: however, it often overloads and over-whelms students with data, making it likely that they will confuse the facts presented
- Hands-on learning, especially in science, has a positive effect on student achievement.
- Teachers who use hands-on learning strategies have students who out-perform their peers on the National Assessment of Educational progress (NAEP) in the areas of science and mathematics.
- Despite the research supporting hands-on activity, it is a fairly uncommon instructional approach.
- Students have higher achievement rates when the focus of instruction is on meaningful conceptualization, especially when it emphasizes their own knowledge of the world.

b=6

-2/3

Build – A – Square- Build-a-square is based on the “Crazy” puzzles where 9 tiles are placed in a 3X3 square arrangement with all edges matching.
- Create 9 tiles with math problems and answers along the edges.
- The puzzle is designed so that the correct formation has all questions and answers matched on the edges.
- Tips: Design the answers for the edges first, then write the specific problems.
- Use more or less squares to tier.
- Add distractors to outside edges and

“letter” pieces at the end.

Nanci Smith

Flippers!

- You will need 2 sheets of construction paper, of different colors. (You’ll only use ½ a sheet of the second color though.)
- Fold the frame color into fourths horizontally (hamburger folds).
- Back-fold the same piece in the opposite directions so that it is well creased and flexible.
- Fold the frame at the center only, and make cuts from the fold up to the next fold line. 7 cuts for 8 sections is easy to do, but cut as many as you like.
- Fold the second color of paper into fourths as well. Cut these apart. You will only use 2 of the strips.
- Basket-weave the two strips into the cut strips of the frame. The two sides need to be woven in opposite directions.
- To use the flipper, write questions on the woven colors. To find the answers, fold the flipper so that the center is pointed at you, then pull the center apart to reveal answer spaces.
- Flipper works in this way on both sides!

Nanci Smith, 2004

Angles Relationship RAFT

** This last entry would take more time than the previous 4 lines, and assesses a little differently. You could offer it as an option with a later due date, but you would need to specify that they need to explain what the angles are, and anything specific that you want to know such as what is the angle’s complement or is there a vertical angle that corresponds, etc.

RAFT Planning Sheet

Know

Understand

Do

How to Differentiate:

- Tiered? (See Equalizer)
- Profile? (Differentiate Format)
- Interest? (Keep options equivalent in learning)
- Other?

Describe it: Look at the subject closely (perhaps with your senses as well as your mind)

Compare it: What is it similar to? What is it different from?

Associate it: What does it make you think of? What comes to your mind when you think of it? Perhaps people? Places? Things? Feelings? Let your mind go and see what feelings you have for the subject.

Analyze it:Tell how it is made? What are it’s traits and attributes?

Apply it: Tell what you can do with it. How can it be used?

Argue for it or against it: Take a stand. Use any kind of reasoning you want – logical, silly, anywhere in between.

Or you can . . . .

Rearrange it

Illustrate it

Question it

Satirize it

Evaluate it

Connect it

Cartoon it

Change it

Solve it

CUBINGArrange ________ into a 3-D collage to show ________

Make a body sculpture to show ________

Create a dance to show

Do a mime to help us understand

Present an interior monologue with dramatic movement that ________

Build/construct a representation of ________

Make a living mobile that shows and balances the elements of ________

Create authentic sound effects to accompany a reading of _______

Show the principle of ________ with a rhythm pattern you create. Explain to us how that works.

Ideas for Cubing in Math

Describe how you would solve ______

Analyze how this problem helps us use mathematical thinking and problem solving

Compare and contrast this problem to one on page _____.

Demonstrate how a professional (or just a regular person) could apply this kink or problem to their work or life.

Change one or more numbers, elements, or signs in the problem. Give a rule for what that change does.

Create an interesting and challenging word problem from the number problem. (Show us how to solve it too.)

Diagram or illustrate the solutionj to the problem. Interpret the visual so we understand it.

Cubing

Cubing

Ideas for CubingCubing

Multiplication Think Dots

It’s easy to remember how to multiply by 0 or 1! Tell how to remember.

Jamie says that multiplying by 10 just adds a 0 to the number. Bryan doesn’t understand this, because any number plus 0 is the same number. Explain what Jamie means, and why her trick can work.

Explain how multiplying by 2 can help with multiplying by 4 and 8. Give at least 3 examples.

We never studied the 7 multiplication facts. Explain why we didn’t need to.

Jorge and his ____ friends each have _____ trading cards. How many trading cards do they have all together? Show the answer to your problem by drawing an array or another picture. Roll a number cube to determine the numbers for each blank.

What is _____ X _____? Find as many ways to show your answer as possible.

- Struggling to Basic Level

Multiplication Think Dots

There are many ways to remember multiplication facts. Start with 0 and go through 10 and tell how to remember how to multiply by each number. For example, how do you remember how to multiply by 0? By 1? By 2? Etc.

There are many patterns in the multiplication chart. One of the patterns deals with pairs of numbers, for example, multiplying by 3 and multiplying by 6 or multiplying by 5 and multiplying by 10. What other pairs of numbers have this same pattern? What is the pattern?

Russell says that 7 X 6 is 42. Kadi says that he can’t know that because we didn’t study the 7 multiplication facts. Russell says he didn’t need to, andhe is right. How might Russell know his answer is correct?

Max says that he can find the answer to a number times 16 simply by knowing the answer to the same number times 2. Explain how Max can figure it out, and give at least two examples.

Alicia and her ____ friends each have _____ necklaces. How many necklaces do they have all together? Show the answer to your problem by drawing an array or another picture. Roll a number cube to determine the numbers for each blank.

What is _____ X _____? Find as many ways to show your answer as possible.

- Middle to High Level

Describe how you would Explain the difference

- solve or roll between adding and
- the die to determine your multiplying fractions,
- own fractions.
- Compare and contrast Create a word problem
- these two problems: that can be solved by
- +
- and (Or roll the fraction die to
- determine your fractions.)
- Describe how people use Model the problem
- fractions every day. ___ + ___ .
- Roll the fraction die to
- determine which fractions
- to add.

Fraction

Think Dots

Nanci Smith

Describe how you would Explain why you need

solve or roll a common denominator

the die to determine your when adding fractions,

own fractions. But not when multiplying.

Can common denominators

Compare and contrast ever be used when dividing

these two problems: fractions?

Create an interesting and challenging word problem

A carpet-layer has 2 yards that can be solved by

of carpet. He needs 4 feet ___ + ____ - ____.

of carpet. What fraction of Roll the fraction die to

his carpet will he use? How determine your fractions.

do you know you are correct?

Diagram and explain the solution to ___ + ___ + ___.

Roll the fraction die to

determine your fractions.

Fraction

Think Dots

Nanci Smith

Designing a Differentiated Learning Contract

- A Learning Contract has the following components
- A Skills Component
- Focus is on skills-based tasks
- Assignments are based on pre-assessment of students’ readiness
- Students work at their own level and pace
- A content component
- Focus is on applying, extending, or enriching key content (ideas, understandings)
- Requires sense making and production
- Assignment is based on readiness or interest
- A Time Line
- Teacher sets completion date and check-in requirements
- Students select order of work (except for required meetings and homework)
- 4. The Agreement
- The teacher agrees to let students have freedom to plan their time
- Students agree to use the time responsibly
- Guidelines for working are spelled out
- Consequences for ineffective use of freedom are delineated
- Signatures of the teacher, student and parent (if appropriate) are placed on the agreement

Differentiating Instruction: Facilitator’s Guide, ASCD, 1997

Date

Student Name

What I will do

What I will use

When I will finish

How I feel about my project How my teacher feels about my project

because

because

Student signature Teacher’s Signature

Learning ContractChapter: _______

Name:______________________

Ck Page/Concept Ck Page/Concept Ck Page/Concept

___ ___________ ___ ___________ ___ ___________

___ ___________ ___ ___________ ___ ___________

___ ___________ ___ ___________ ___ ___________

___ ___________ ___ ___________ ___ ___________

Enrichment Options: ______________________________________________ Special Instructor

______________________________ _____ _____ _____ _____ _____ _____ _____

______________________________ _____ _____ _____ _____ _____ _____ _____

______________________________ _____ _____ _____ _____ _____ _____ _____

Your Idea:

______________________________ _____ _____ _____ _____ _____ _____ _____

Working Conditions

________________________________________________________________________

________________________________________________________________________

_________________________________ ___________________________________

Teacher’s signature Student’s signature

Key Skills: Graphing and Measuring

Key Concepts: Relative Sizes

Note to User: This is a Grade 3 math contract for students below grade level in these skills

Find a friend and do

Board math with

Problems 1-10 on

Page 71 of our

Math book.

Remember the

“no more than 4”

rule

Come to the red

math workshop

on Monday

and

Tuesday

Use the dominoes to

solve the problems in

your folder. Draw

and then write

your answers

Work at the measuring and graphing center until you complete the red work

Solve the

great graph

mystery in your

math folder. Check

Your answers with a

buddy, then with the teacher

Design an

animal on graph

paper using the

creature blueprint.

Get your graph approved. Then make a drawing, painting, or model of it

Key Skills: Graphing and Measuring

Key Concepts: Relative Sizes

Note to User:This is a Grade 3 math contract for students at or near grade level in these skills

Come to the green math workshop

on Monday

and

Friday

Work the even numbered

problems on page 71

of our math book. Use

the export of the day

to audit your

work

Complete the dominoes multiplication challenge. Record your answers on the wall chart

Work at the measuring and graphing center until you complete the green work

Solve the

great graph

mystery in your

math folder. You can

Work with someone on

The green team if you’d

Like. Check your answer

With the teacher

Design an

animal on graph

paper using the

creature blueprint.

Get your graph approved. Then make a drawing, painting, or model of it

Key Skills: Graphing and Measuring

Key Concepts: Relative Sizes

Note to User: This is a Grade 3 math contract for students advanced in these skills

Come to the blue

math workshop

on

Tuesday or Thursday morning

Complete the

extension problems

on graphing on page

74 of our math book.

Use a peer

monitor to

audit your

work.

Do a timed test of

Two-digit multiplication. Use a peer monitor

Work at the measuring and graphing center until you complete the blue work

Solve the

graph mystery

in your folder. You

can work with someone

on the blue team if you’d

like.

Find a

place in

our school to

make a pattern

graph of. Make the

graph and create three

problems for a classmate

to solve.

Proportional Reasoning Think-Tac-Toe

Directions: Choose one option in each row to complete. Check the box of the choice you make, and turn this page in with your finished selections.

Nanci Smith, 2004

Use this template to help you plan a menu for your classroom

Menu: ____________________

Due: All items in the main dish and the specified number of side dishes must be completed by the due date. You may select among the side dishes and you may decide to do some of the dessert items as well.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Main Dish (complete all)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Side Dish (select ____)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Dessert

Winning Strategies for Classroom Management

Similar Figures Menu

Imperatives (Do all 3):

- Write a mathematical definition of “Similar Figures.” It must include all pertinent vocabulary, address all concepts and be written so that a fifth grade student would be able to understand it. Diagrams can be used to illustrate your definition.
- Generate a list of applications for similar figures, and similarity in general. Be sure to think beyond “find a missing side…”
- Develop a lesson to teach third grade students who are just beginning to think about similarity.

Similar Figures Menu

Negotiables (Choose 1):

- Create a book of similar figure applications and problems. This must include at least 10 problems. They can be problems you have made up or found in books, but at least 3 must be application problems. Solver each of the problems and include an explanation as to why your solution is correct.
- Show at least 5 different application of similar figures in the real world, and make them into math problems. Solve each of the problems and explain the role of similarity. Justify why the solutions are correct.

Similar Figures Menu

Optionals:

- Create an art project based on similarity. Write a cover sheet describing the use of similarity and how it affects the quality of the art.
- Make a photo album showing the use of similar figures in the world around us. Use captions to explain the similarity in each picture.
- Write a story about similar figures in a world without similarity.
- Write a song about the beauty and mathematics of similar figures.
- Create a “how-to” or book about finding and creating similar figures.

OPTIONS FOR DIFFERENTIATION OF INSTRUCTION

To Differentiate Instruction By Readiness

To Differentiate Instruction By Interest

To Differentiate Instruction by Learning Profile

CA Tomlinson, UVa ‘97

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