Transcript Document

Differentiating Mathematics in the
Elementary Classroom
Raising Student Achievement Conference
St. Charles, IL
December 4, 2007
"In the end, all learners need your
energy, your heart and your mind.
They have that in common because
they are young humans. How they
need you however, differs. Unless
we understand and respond to
those differences, we fail many
learners." *
* Tomlinson, C.A. (2001). How to differentiate instruction in mixed ability
classrooms (2nd Ed.). Alexandria, VA: ASCD.
Nanci Smith
Educational Consultant
Curriculum and Professional Development
Cave Creek, AZ
[email protected]
Differentiation of Instruction
Is a teacher’s response to learner’s needs
guided by general principles of differentiation
Respectful tasks
Flexible grouping
Continual assessment
Teachers Can Differentiate Through:
Content
Process
Product
According to Students’
Readiness
Interest
Learning Profile
What’s the point of differentiating
in these different ways?
Readiness
Interest
Learning
Profile
Growth
Motivation
Efficiency
Key Principles of a
Differentiated Classroom
• The teacher understands, appreciates,
and builds upon student differences.
Source: Tomlinson, C. (2000). Differentiating Instruction for Academic Diversity. San Antonio, TX: ASCD
READINESS
What does READINESS mean?
It is the student’s entry point
relative to a particular
understanding or skill.
C.A.Tomlinson, 1999
A Few Routes to READINESS
DIFFERENTIATION
Varied texts by reading level
Varied supplementary materials
Varied scaffolding
• reading
• writing
• research
• technology
Tiered tasks and procedures
Flexible time use
Small group instruction
Homework options
Tiered or scaffolded assemssment
Compacting
Mentorships
Negotiated criteria for quality
Varied graphic organizers
For example…
Providing support
needed for a student
to succeed in work
slightly beyond his/her
comfort zone.
•Directions that give more structure – or less
•Tape recorders to help with reading or writing beyond the student’s grasp
•Icons to help interpret print
•Reteaching / extending teaching
•Modeling
•Clear criteria for success
•Reading buddies (with appropriate directions)
•Double entry journals with appropriate challenge
•Teaching through multiple modes
•Use of manipulatives when needed
•Gearing reading materials to student reading level
•Use of study guides
•Use of organizers
•New American Lecture
Tomlinson, 2000
Compacting
1.
2.
3.
4.
5.
Identify the learning objectives or standards ALL students must
learn.
Offer a pretest opportunity OR plan an alternate path through
the content for those students who can learn the required
material in less time than their age peers.
Plan and offer meaningful curriculum extensions for kids who
qualify.
**Depth and Complexity
Applications of the skill being taught
Learning Profile tasks based on understanding the
process instead of skill practice
Differing perspectives, ideas across time, thinking
like a mathematician
**Orbitals and Independent studies.
Eliminate all drill, practice, review, or preparation for students
who have already mastered such things.
Keep accurate records of students’ compacting activities:
document mastery.
Strategy: Compacting
Developing a Tiered Activity
1
Select the activity organizer
•concept
Essential to building
•generalization
a framework of
2
• readiness range
• interests
• learning profile
• talents
understanding
3
Create an activity that is
• interesting
• high level
• causes students to use
key skill(s) to understand
a key idea
Think about your students/use assessments
skills
reading
thinking
information
4
Chart the
complexity of
the activity
High skill/
Complexity
Low skill/
complexity
5
Clone the activity along the ladder as
needed to ensure challenge and success
for your students, in
•
materials – basic to advanced
•
•
•
form of expression – from familiar to
unfamiliar
from personal experience to removed
from personal experience
equalizer
6
Match task to student based on
student profile and task
requirements
The Equalizer
1. Foundational
Transformational
Information, Ideas, Materials, Applications
2. Concrete
Abstract
Representations, Ideas, Applications, Materials
1. Simple
Complex
Resources, Research, Issues, Problems, Skills, Goals
2. Single Facet
Multiple Facets
Directions, Problems, Application, Solutions, Approaches, Disciplinary Connections
3. Small Leap
Great Leap
Application, Insight, Transfer
4. More Structured
More Open
Solutions, Decisions, Approaches
5. Less Independence
Greater Independence
Planning, Designing, Monitoring
6. Slow
Pace of Study, Pace of Thought
Quick
Adding Fractions
Green Group
Use Cuisinaire rods or fraction
circles to model simple fraction
addition problems. Begin with
common denominators and work
up to denominators with common
factors such as 3 and 6.
Explain the pitfalls and hurrahs of
adding fractions by making a
picture book.
Red Group
Use Venn diagrams to model LCMs
(least common multiple). Explain
how this process can be used to
find common denominators. Use
the method on more challenging
addition problems.
Write a manual on how to add
fractions. It must include why a
common denominator is needed,
and at least three ways to find it.
Blue Group
Manipulatives such as Cuisinaire
rods and fraction circles will be
available as a resource for the
group. Students use factor trees
and lists of multiples to find
common denominators. Using this
approach, pairs and triplets of
fractions are rewritten using
common denominators. End by
adding several different problems
of increasing challenge and length.
Suzie says that adding fractions is
like a game: you just need to know
the rules. Write game instructions
explaining the rules of adding
fractions.
BRAIN RESEARCH SHOWS THAT. . .
Eric Jensen, Teaching With the Brain in Mind, 1998
Choices
vs.
Required
content, process, product
groups, resources environment
Relevant
no student voice
restricted resources
vs.
Irrelevant
meaningful
connected to learner
impersonal
out of context
deep understanding
Engaging
only to pass a test
vs.
emotional, energetic
hands on, learner input
Passive
low interaction
lecture seatwork
EQUALS
Increased intrinsic
MOTIVATION
Increased
APATHY & RESENTMENT
-CHOICE-
The Great Motivator!
• Requires children to be aware of their own readiness, interests, and
learning profiles.
• Students have choices provided by the teacher. (YOU are still in
charge of crafting challenging opportunities for all kiddos – NO
taking the easy way out!)
• Use choice across the curriculum: writing topics, content writing
prompts, self-selected reading, contract menus, math problems,
spelling words, product and assessment options, seating, group
arrangement, ETC . . .
• GUARANTEES BUY-IN AND ENTHUSIASM FOR LEARNING!
• Research currently suggests that CHOICE should be offered 35%
of the time!!
Assessments
The assessments used in this learning profile
section can be downloaded at:
www.e2c2.com/fileupload.asp
Download the file entitled “Profile
Assessments for Cards.”
How Do You Like to Learn?
1. I study best when it is quiet.
2. I am able to ignore the noise of
other people talking while I am working.
3. I like to work at a table or desk.
4. I like to work on the floor.
5. I work hard by myself.
6. I work hard for my parents or teacher.
7. I will work on an assignment until it is completed, no
matter what.
8. Sometimes I get frustrated with my work
and do not finish it.
9. When my teacher gives an assignment, I like to
have exact steps on how to complete it.
10. When my teacher gives an assignment, I like to
create my own steps on how to complete it.
11. I like to work by myself.
12. I like to work in pairs or in groups.
13. I like to have unlimited amount of time to work on
an assignment.
14. I like to have a certain amount of time to work on
an assignment.
15. I like to learn by moving and doing.
16. I like to learn while sitting at my desk.
Yes No
Yes
Yes
Yes
Yes
Yes
No
No
No
No
No
Yes No
Yes No
Yes No
Yes No
Yes No
Yes No
Yes No
Yes No
Yes No
Yes No
My Way
An expression Style Inventory
K.E. Kettle J.S. Renzull, M.G. Rizza
University of Connecticut
Products provide students and professionals with a way to express what they
have learned to an audience. This survey will help determine the kinds of
products YOU are interested in creating.
My Name is: ____________________________________________________
Instructions:
Read each statement and circle the number that shows to what extent YOU are
interested in creating that type of product. (Do not worry if you are unsure of how
to make the product).
Not At All Interested
Of Little Interest
Moderately Interested
Interested
Very Interested
1. Writing Stories
1
2
3
4
5
2. Discussing what I
have learned
1
2
3
4
5
3. Painting a picture
1
2
3
4
5
4. Designing a
computer software
project
1
2
3
4
5
5. Filming & editing a
video
1
2
3
4
5
6. Creating a company
1
2
3
4
5
7. Helping in the
community
1
2
3
4
5
8. Acting in a play
1
2
3
4
5
Not At All Interested
Of Little Interest
Moderately Interested
Interested
Very Interested
9. Building an
invention
1
2
3
4
5
10. Playing musical
instrument
1
2
3
4
5
11. Writing for a
newspaper
1
2
3
4
5
12. Discussing ideas
1
2
3
4
5
13. Drawing pictures
for a book
1
2
3
4
5
14. Designing an
interactive computer
project
1
2
3
4
5
15. Filming & editing
a television show
1
2
3
4
5
16. Operating a
business
1
2
3
4
5
17. Working to help
others
1
2
3
4
5
18. Acting out an
event
1
2
3
4
5
19. Building a project
1
2
3
4
5
20. Playing in a band
1
2
3
4
5
21. Writing for a
magazine
1
2
3
4
5
22. Talking about my
project
1
2
3
4
5
23. Making a clay
sculpture of a
character
1
2
3
4
5
Not At All Interested
Of Little Interest
Moderately Interested
Interested
Very Interested
24. Designing
information for the
computer internet
1
2
3
4
5
25. Filming & editing
a movie
1
2
3
4
5
26. Marketing a
product
1
2
3
4
5
27. Helping others by
supporting a social
cause
1
2
3
4
5
28. Acting out a story
1
2
3
4
5
29. Repairing a
machine
1
2
3
4
5
30. Composing music
1
2
3
4
5
31. Writing an essay
1
2
3
4
5
32. Discussing my
research
1
2
3
4
5
33. Painting a mural
1
2
3
4
5
34. Designing a
computer
1
2
3
4
5
35. Recording &
editing a radio show
1
2
3
4
5
36. Marketing an idea
1
2
3
4
5
37. Helping others by
fundraising
1
2
3
4
5
38. Performing a skit
1
2
3
4
5
Not At All Interested
Of Little Interest
Moderately Interested
Interested
Very Interested
39. Constructing a
working model.
1
2
3
4
5
40. Performing music
1
2
3
4
5
41. Writing a report
1
2
3
4
5
42. Talking about my
experiences
1
2
3
4
5
43. Making a clay
sculpture of a scene
1
2
3
4
5
44. Designing a multimedia computer show
1
2
3
4
5
45. Selecting slides
and music for a slide
show
1
2
3
4
5
46. Managing
investments
1
2
3
4
5
47. Collecting
clothing or food to
help others
1
2
3
4
5
48. Role-playing a
character
1
2
3
4
5
49. Assembling a kit
1
2
3
4
5
50. Playing in an
orchestra
1
2
3
4
5
Instructions: My
Way …A Profile
Write your score
beside each
number. Add each
Row to determine
your expression
style profile.
Products
Written
Oral
Artistic
Computer
Audio/Visual
Commercial
Service
Dramatization
Manipulative
Musical
1. ___
2. ___
3. ___
4. ___
5. ___
6. ___
7. ___
8. ___
9. ___
10.___
11.
12.
13.
14.
15.
16.
77.
18.
19.
20.
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___
___
___
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___
___
21. ___
22. ___
23. ___
24. ___
25. ___
26. ___
27. ___
28. ___
29. ___
30 . ___
31. ___
32. ___
33. ___
34. ___
35. ___
36. ___
37. ___
38. ___
39. ___
40. ___
41. ___
42. ___
43. ___
44. ___
45. ___
46. ___
47. ___
48. ___
49. ___
50. ___
Total
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
Learner Profile Card
Gender Stripe
Auditory, Visual, Kinesthetic
Analytical, Creative, Practical
Modality
Sternberg
Student’s
Interests
Multiple Intelligence Preference
Gardner
Array
Inventory
Nanci Smith,Scottsdale,AZ
Differentiation Using
LEARNING PROFILE
• Learning profile refers to how an
individual learns best - most efficiently
and effectively.
• Teachers and their students may
differ in learning profile preferences.
Learning Profile Factors
Group Orientation
independent/self orientation
group/peer orientation
adult orientation
combination
Learning Environment
Gender
&
Culture
Cognitive Style
Creative/conforming
Essence/facts
Expressive/controlled
Nonlinear/linear
Inductive/deductive
People-oriented/task or Object oriented
Concrete/abstract
Collaboration/competition
Interpersonal/introspective
Easily distracted/long Attention span
Group achievement/personal achievement
Oral/visual/kinesthetic
Reflective/action-oriented
quiet/noise
warm/cool
still/mobile
flexible/fixed
“busy”/”spare”
Intelligence Preference
analytic
practical
creative
verbal/linguistic
logical/mathematical
spatial/visual
bodily/kinesthetic
musical/rhythmic
interpersonal
intrapersonal
naturalist
existential
Activity 2.5 – The Modality Preferences Instrument (HBL, p. 23)
Follow the directions below to get a score that will indicate your own modality (sense) preference(s). This instrument, keep
in mind that sensory preferences are usually evident only during prolonged and complex learning tasks.
Identifying Sensory Preferences
Directions: For each item, circle “A” if you agree that the statement describes you most of the time. Circle “D” if you
disagree that the statement describes you most of the time.
1.
I Prefer reading a story rather than listening to someone tell it.
A
D
2.
I would rather watch television than listen to the radio.
A
D
3.
I remember faces better than names.
A
D
4.
I like classrooms with lots of posters and pictures around the room.
A
D
5.
The appearance of my handwriting is important to me.
A
D
6.
I think more often in pictures.
A
D
7.
I am distracted by visual disorder or movement.
A
D
8.
I have difficulty remembering directions that were told to me.
A
D
9.
I would rather watch athletic events than participate in them.
A
D
10.
I tend to organize my thoughts by writing them down.
A
D
11.
My facial expression is a good indicator of my emotions.
A
D
12.
I tend to remember names better than faces.
A
D
13.
I would enjoy taking part in dramatic events like plays.
A
D
14.
I tend to sub vocalize and think in sounds.
A
D
15.
I am easily distracted by sounds.
A
D
16.
I easily forget what I read unless I talk about it.
A
D
17.
I would rather listen to the radio than watch TV
A
D
18.
My handwriting is not very good.
A
D
19.
When faced with a problem , I tend to talk it through.
A
D
20.
I express my emotions verbally.
A
D
21.
I would rather be in a group discussion than read about a topic.
A
D
22.
I prefer talking on the phone rather than writing a letter to someone.
A
D
23.
I would rather participate in athletic events than watch them.
A
D
24.
I prefer going to museums where I can touch the exhibits.
A
D
25.
My handwriting deteriorates when the space becomes smaller.
A
D
26.
My mental pictures are usually accompanied by movement.
A
D
27.
I like being outdoors and doing things like biking, camping, swimming, hiking etc.
A D
28.
I remember best what was done rather then what was seen or talked about.
A
D
29.
When faced with a problem, I often select the solution involving the greatest activity.
A
D
30.
I like to make models or other hand crafted items.
A
D
31.
I would rather do experiments rather then read about them.
A
D
32.
My body language is a good indicator of my emotions.
A
D
33.
I have difficulty remembering verbal directions if I have not done the activity before.
A
D
Interpreting the Instrument’s Score
Total the number of “A” responses in items 1-11
_____
This is your visual score
Total the number of “A” responses in items 12-22
_____
This is your auditory score
Total the number of “A” responses in items 23-33
_____
This is you tactile/kinesthetic score
If you scored a lot higher in any one area: This indicates that this modality is very probably your preference during a protracted and complex
learning situation.
If you scored a lot lower in any one area: This indicates that this modality is not likely to be your preference(s) in a learning situation.
If you got similar scores in all three areas: This indicates that you can learn things in almost any way they are presented.
Multiplication Facts: 4’s and 8’s
•
Visual:
–
•
Make two posters - one will diagram all of the 4
multiplication facts and the other diagrams the 8
multiplication facts.
Auditory:
–
•
Put together a skit or newscast about multiplying by 4 and
8. Have lots of examples!
Kinesthetic:
–
–
Play multiplication rummy or memory
Use counters to model the 4 and 8 multiplication facts.
List all of the resulting equations and answers.
Parallel Lines Cut by a Transversal
• Visual: Make posters showing all the angle
relations formed by a pair of parallel lines
cut by a transversal. Be sure to color code
definitions and angles, and state the
relationships between all possible angles.
2
1
4
3
5
6
8
7
Smith & Smarr, 2005
Parallel Lines Cut by a Transversal
• Auditory: Play “Shout Out!!” Given the
diagram below and commands on strips of paper
(with correct answers provided), players take turns
being the leader to read a command. The first
player to shout out a correct answer to the
command, receives a point. The next player
becomes the next leader. Possible commands:
– Name an angle supplementary
supplementary to angle 1.
– Name an angle congruent
to angle 2.
2
6
5
1
4
3
8
7
Smith & Smarr, 2005
Parallel Lines Cut by a Transversal
• Kinesthetic: Walk It
Tape the diagram below
on the floor with masking
tape. Two players stand in
assigned angles. As a
team, they have to tell
what they are called (ie:
vertical angles) and their
relationships (ie:
congruent). Use all angle
combinations, even if
there is not a name or
relationship. (ie: 2 and 7)
1
2
5
3
4
8
6
7
Smith & Smarr, 2005
EIGHT STYLES OF LEARNING
TYPE
CHARACTERISTICS
LIKES TO
IS GOOD AT
LEARNS BEST BY
LINGUISTIC
LEARNER
Learns through the
manipulation of words. Loves
to read and write in order to
explain themselves. They also
tend to enjoy talking
Read
Write
Tell stories
Memorizing
names, places,
dates and trivia
Saying, hearing and
seeing words
“The Questioner”
Looks for patterns when
solving problems. Creates a set
of standards and follows them
when researching in a
sequential manner.
Do experiments
Figure things out
Work with numbers
Ask questions
Explore patterns and
relationships
Math
Reasoning
Logic
Problem solving
Categorizing
Classifying
Working with abstract
patterns/relationships
SPATIAL
LEARNER
Learns through pictures, charts,
graphs, diagrams, and art.
Draw, build, design
and create things
Daydream
Look at pictures/slides
Watch movies
Play with machines
Imagining things
Sensing changes
Mazes/puzzles
Reading maps,
charts
Visualizing
Dreaming
Using the mind’s eye
Working with
colors/pictures
Learning is often easier for
these students when set to
music or rhythm
Sing, hum tunes
Listen to music
Play an instrument
Respond to music
Picking up sounds
Remembering
melodies
Noticing pitches/
rhythms
Keeping time
Rhythm
Melody
Music
“The Word Player”
LOGICAL/
Mathematical
Learner
“The Visualizer”
MUSICAL
LEARNER
“The Music
Lover”
EIGHT STYLES OF LEARNING, Cont’d
TYPE
CHARACTERISTICS
LIKES TO
IS GOOD AT
LEARNS BEST BY
BODILY/
Kinesthetic
Learner
Eager to solve problems
physically. Often doesn’t read
directions but just starts on a
project
Move around
Touch and talk
Use body
language
Physical activities
(Sports/dance/
acting)
crafts
Touching
Moving
Interacting with space
Processing knowledge
through bodily sensations
Likes group work and
working cooperatively to
solve problems. Has an
interest in their community.
Have lots of
friends
Talk to people
Join groups
Understanding people
Leading others
Organizing
Communicating
Manipulating
Mediating conflicts
Sharing
Comparing
Relating
Cooperating
interviewing
Enjoys the opportunity to
reflect and work
independently. Often quiet
and would rather work on
his/her own than in a group.
Work alone
Pursue own
interests
Understanding self
Focusing inward on
feelings/dreams
Pursuing interests/
goals
Being original
Working along
Individualized projects
Self-paced instruction
Having own space
Enjoys relating things to their
environment. Have a strong
connection to nature.
Physically
experience nature
Do observations
Responds to
patterning nature
Exploring natural
phenomenon
Seeing connections
Seeing patterns
Reflective Thinking
Doing observations
Recording events in Nature
Working in pairs
Doing long term projects
“The Mover”
INTERpersonal
Learner
“The Socializer”
INTRApersonal
Learner
“The Individual”
NATURALIST
“The Nature
Lover”
Multiplying by 3 and 6!
•
•
•
•
•
•
Play Multiplication Memory card game
(Kinesthetic, interpersonal).
Make a picture book of multiplication facts for 3
and/or 6 (visual/spatial).
Make up a song about (or of) the multiplication
facts for 3 and/or 6 (musical).
Write a diary entry about the 3 and 6
multiplication facts. What are they? How can
you remember them? If you forget one, how
could you figure it out? (Intrapersonal / verbal
linguistic)
Write a story that involves multiplication by 3
and 6 (verbal linguistic).
Show as many different models of multiplication
by 3 and 6 of which you can think. How is
multiplying by 6 related to multiplying by 3?
(Logical / Mathematical)
Sternberg’s Three Intelligences
Creative
Analytical
Practical
•We all have some of each of these intelligences, but are usually
stronger in one or two areas than in others.
•We should strive to develop as fully each of these intelligences
in students…
• …but also recognize where students’ strengths lie and teach
through those intelligences as often as possible, particularly
when introducing new ideas.
Thinking About the Sternberg Intelligences
ANALYTICAL
Linear – Schoolhouse Smart - Sequential
Show the parts of _________ and how they work.
Explain why _______ works the way it does.
Diagram how __________ affects __________________.
Identify the key parts of _____________________.
Present a step-by-step approach to _________________.
PRACTICAL
Streetsmart – Contextual – Focus on Use
Demonstrate how someone uses ________ in their life or work.
Show how we could apply _____ to solve this real life problem ____.
Based on your own experience, explain how _____ can be used.
Here’s a problem at school, ________. Using your knowledge of
______________, develop a plan to address the problem.
CREATIVE
Innovator – Outside the Box – What If - Improver
Find a new way to show _____________.
Use unusual materials to explain ________________.
Use humor to show ____________________.
Explain (show) a new and better way to ____________.
Make connections between _____ and _____ to help us understand ____________.
Become a ____ and use your “new” perspectives to help us think about
____________.
Triarchic Theory of Intelligences
Robert Sternberg
Mark each sentence T if you like to do the activity and F if you do not like to do
the activity.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Analyzing characters when I’m reading or listening to a story
Designing new things
Taking things apart and fixing them
Comparing and contrasting points of view
Coming up with ideas
Learning through hands-on activities
Criticizing my own and other kids’ work
Using my imagination
Putting into practice things I learned
Thinking clearly and analytically
Thinking of alternative solutions
Working with people in teams or groups
Solving logical problems
Noticing things others often ignore
Resolving conflicts
___
___
___
___
___
___
___
___
___
___
___
___
___
___
___
Triarchic Theory of Intelligences
Robert Sternberg
Mark each sentence T if you like to do the activity and F if you do not like to do
the activity.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
Evaluating my own and other’s points of view
Thinking in pictures and images
Advising friends on their problems
Explaining difficult ideas or problems to others
Supposing things were different
Convincing someone to do something
Making inferences and deriving conclusions
Drawing
Learning by interacting with others
Sorting and classifying
Inventing new words, games, approaches
Applying my knowledge
Using graphic organizers or images to organize your thoughts
Composing
Adapting to new situations
___
___
___
___
___
___
___
___
___
___
___
___
___
___
___
Triarchic Theory of Intelligences – Key
Robert Sternberg
Transfer your answers from the survey to the key. The column with the most True
responses is your dominant intelligence.
Analytical
1. ___
4. ___
7. ___
10. ___
13. ___
16. ___
19. ___
22. ___
25. ___
28. ___
Creative
2. ___
5. ___
8. ___
11. ___
14. ___
17. ___
20. ___
23. ___
26. ___
29. ___
Practical
3. ___
6. ___
9. ___
12. ___
15. ___
18. ___
21. ___
24. ___
27. ___
30. ___
Total Number of True:
Analytical ____
Creative _____
Practical _____
• Analytical
– Draw arrays for multiplying by 3 and 6, and list the facts next to
each array. Next, make a list of as many patterns as you can find
from the multiplication facts. Make a poster to help the class
remember the 3 and 6 multiplication facts.
• Practical
– You and 5 of your friends go to the zoo. You all pay the
admission of $3.00 each. You each buy a box lunch and each
lunch costs $5.00. Three of your friends decide to buy a stuffed
animal at the gift shop. The stuffed animals each cost $7.00.
– How much money was spent on admission in total?
– How much money was spent on lunch in total?
– How much money was spent on stuffed animals in total?
– How much money was spent in total?
– Show how you know each of your answers is correct by explaining
or drawing how you found each answer.
• Creative
– Complete the following RAFT
ROLE
AUDIENCE
FORMAT
TOPIC
Multiplication by 3 Multiplication by 2
Friendly letter
If someone knows
you, they can find
me.
Multiplication by 3 Multiplication by 6
Friendly letter
If someone knows
you, they can find
me.
OR:
– Think of a way to remember the 3 and/or 6 multiplication
facts. Make a poster, explain, sing or draw how to remember
them.
Understanding Order of Operations
Analytic Task
Make a chart that shows all ways you
can think of to use order of operations
to equal 18.
Practical Task
A friend is convinced that order of
operations do not matter in math. Think
of as many ways to convince your friend
that without using them, you won’t
necessarily get the correct answers!
Give lots of examples.
Creative Task
Write a book of riddles that involve order
of operations. Show the solution and
pictures on the page that follows each
riddle.
Key Principles of a
Differentiated Classroom
• Assessment and instruction are
inseparable.
Source: Tomlinson, C. (2000). Differentiating Instruction for Academic Diversity. San Antonio, TX: ASCD
Pre-Assessment
• What the student already knows about what is
being planned
• What standards, objectives, concepts & skills the
individual student understands
• What further instruction and opportunities for
mastery are needed
• What requires reteaching or enhancement
• What areas of interests and feelings are in the
different areas of the study
• How to set up flexible groups: Whole, individual,
partner, or small group
THINKING ABOUT
ON-GOING ASSESSMENT
STUDENT DATA SOURCES
1. Journal entry
2. Short answer test
3. Open response test
4. Home learning
5. Notebook
6. Oral response
7. Portfolio entry
8. Exhibition
9. Culminating product
10. Question writing
11. Problem solving
TEACHER DATA
MECHANISMS
1. Anecdotal records
2. Observation by checklist
3. Skills checklist
4. Class discussion
5. Small group interaction
6. Teacher – student
conference
7. Assessment stations
8. Exit cards
9. Problem posing
10. Performance tasks and
rubrics
Key Principles of a
Differentiated Classroom
• The teacher adjusts content, process,
and product in response to student
readiness, interests, and learning
profile.
Source: Tomlinson, C. (2000). Differentiating Instruction for Academic Diversity. San Antonio, TX: ASCD
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.
Make Card Games!
Make Card Games!
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.
m=3
• Add distractors to outside edges and
b=6
-2/3
“letter” pieces at the end.
Nanci Smith
The ROLE of writer, speaker,
artist, historian, etc.
RAFT
An AUDIENCE of fellow writers,
students, citizens, characters, etc.
Through a FORMAT that is
written, spoken, drawn, acted, etc.
electron
neutron
proton
A TOPIC related to curriculum
content in greater depth.
RAFT ACTIVITY ON FRACTIONS
Role
Audience
Format
Topic
Fraction
Whole Number
Petitions
To be considered Part of the
Family
Improper Fraction
Mixed Numbers
Reconciliation Letter
Were More Alike than
Different
A Simplified Fraction
A Non-Simplified Fraction
Public Service
Announcement
A Case for Simplicity
Greatest Common Factor
Common Factor
Nursery Rhyme
I’m the Greatest!
Equivalent Fractions
Non Equivalent
Personal Ad
How to Find Your Soul Mate
Least Common Factor
Multiple Sets of Numbers
Recipe
The Smaller the Better
Like Denominators in an
Additional Problem
Unlike Denominators in an
Addition Problem
Application form
To Become A Like
Denominator
A Mixed Number that
Needs to be Renamed to
Subtract
5th Grade Math Students
Riddle
What’s My New Name
Like Denominators in a
Subtraction Problem
Unlike Denominators in a
Subtraction Problem
Story Board
How to Become a Like
Denominator
Fraction
Baker
Directions
To Double the Recipe
Estimated Sum
Fractions/Mixed Numbers
Advice Column
To Become Well Rounded
Angles Relationship RAFT
Role
Audience
Format
Topic
One vertical angle
Opposite vertical angle
Poem
It’s like looking in a mirror
Interior (exterior) angle
Alternate interior (exterior)
angle
Invitation to a family
reunion
My separated twin
Acute angle
Missing angle
Wanted poster
Wanted: My complement
An angle less than 180
Supplementary
angle
Persuasive speech
Together, we’re a straight angle
**Angles
Humans
Video
See, we’re everywhere!
** 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.
Algebra RAFT
Role
Audience
Format
Topic
Coefficient
Variable
Email
We belong together
Scale / Balance
Students
Advice column
Variable
Humans
Monologue
All that I can be
Variable
Algebra students
Instruction manual
How and why to
isolate me
Algebra
Public
Passionate plea
Why you really do
need me!
Keep me in mind
when solving an
equation
RAFT Planning Sheet
Know
Understand
Do
How to Differentiate:
• Tiered? (See Equalizer)
• Profile? (Differentiate Format)
• Interest? (Keep options equivalent in
learning)
• Other?
Role
Audience
Format
Topic
Cubing
Cubing
Ideas for Cubing
•
•
•
•
•
•
•
•
•
Arrange ________ 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.
Cubing
•
•
•
•
•
•
•
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.
Multiplication Think Dots
• Struggling to Basic Level
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.
Multiplication Think Dots
• Middle to High Level
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, and he 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.
Describe how you would
1 3

5 5
solve
or roll
the die to determine your
Explain the difference
between adding and
multiplying fractions,
own fractions.
Compare and contrast
Create a word problem
these two problems:
that can be solved by
+
and
1 1

3 2
Nanci Smith
1 2 11
 
3 5 15
(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.
Nanci Smith
Describe how you would
solve
2 3 1
 
13 7 91
or roll
Explain why you need
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?
1 1
3 1
 and 
3 2
7 7
Create an interesting and
challenging word problem
Nanci Smith
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.
Designing a Differentiated Learning
Contract
A Learning Contract has the following components
1. 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
2. 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
3. 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
Personal Agenda
Montgomery
County, MD
Personal Agenda for _______________________________________
Starting Date _____________________________________________________
Teacher & student
initials at
completion
Task
Special
Instructions
Remember to complete your daily planning log; I’ll call on you for
conferences & instructions.
Multiplication Think Dots
• Struggling to Basic Level
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.
Multiplication Think Dots
• Middle to High Level
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, and he 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.
Describe how you would
1 3

5 5
solve
or roll
the die to determine your
Explain the difference
between adding and
multiplying fractions,
own fractions.
Compare and contrast
Create a word problem
these two problems:
that can be solved by
+
and
1 1

3 2
Nanci Smith
1 2 11
 
3 5 15
(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.
Nanci Smith
Describe how you would
solve
2 3 1
 
13 7 91
or roll
Explain why you need
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?
1 1
3 1
 and 
3 2
7 7
Create an interesting and
challenging word problem
Nanci Smith
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.
Level 1:
1. a, b, c and d each represent a different value. If a = 2, find
b, c, and d.
a+b=c
a–c=d
a+b=5
2. Explain the mathematical reasoning involved in solving
card 1.
3. Explain in words what the equation 2x + 4 = 10 means.
Solve the problem.
4. Create an interesting word problem that is modeled by
8x – 2 = 7x.
5. Diagram how to solve 2x = 8.
6. Explain what changing the “3” in 3x = 9 to a “2” does to
the value of x. Why is this true?
Level 2:
1. a, b, c and d each represent a different value. If a = -1, find
b, c,
and d.
a+b=c
b+b=d
c – a = -a
2. Explain the mathematical reasoning involved in solving
card 1.
3. Explain how a variable is used to solve word problems.
4. Create an interesting word problem that is modeled by
2x + 4 = 4x – 10. Solve the problem.
5. Diagram how to solve 3x + 1 = 10.
6. Explain why x = 4 in 2x = 8, but x = 16 in ½ x = 8. Why
does this make sense?
Level 3:
1. a, b, c and d each represent a different value. If a = 4, find
b,
c, and d.
a+c=b
b-a=c
cd = -d
d+d=a
2. Explain the mathematical reasoning involved in solving
card 1.
3. Explain the role of a variable in mathematics. Give examples.
4. Create an interesting word problem that is modeled by
3x  1  5x  7. Solve the problem.
5. Diagram how to solve 3x + 4 = x + 12.
6. Given ax = 15, explain how x is changed if a is large or a is
small in value.
Learning Contracts
Contracts take a number of forms that begin
with an agreement between student and
teacher.
The teacher grants certain freedoms and
choices about how a student will complete
tasks, and the student agrees to use the
freedoms appropriately in designing and
completing work according to specifications.
Strategy: Learning Contracts
Designing a Differentiated Learning
Contract
A Learning Contract has the following components
1. 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
2. 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
3. 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
Personal Agenda
Montgomery
County, MD
Personal Agenda for _______________________________________
Starting Date _____________________________________________________
Teacher & student
initials at
completion
Task
Special
Instructions
Remember to complete your daily planning log; I’ll call on you for
conferences & instructions.
Personal Agenda
Agenda for:___________
Starting Date: ___________
TASK
• Complete Hypercard stack
showing how a volcano
works
• Read your personal choice
biography
• Practice adding fraction
by completing number
problems & word
problems on pp 101-106
of workbook
Special Instructions
• Be sure to show scientific
accuracy & computer skill
• Keep a reading log of your
progress
• Come to the teacher or a
friend for help if you get
stuck
Work Log
Date Goal
Actual
The Red Contract
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
Read
Apply
Extend
How big
is a foot?
Work with a
friend to graph
the size of at
least 6 things
on the list of
“10 terrific
things.” Label
each thing with
how you know
the size
Make a
group story
or one of
your own –
that uses
measuremen
t and at least
one graph.
Turn it into a
book at the
author center
The Green Contract
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
Read
Apply
Extend
Alexande
r Who
Used to
be Rich
Last
Sunday or
Ten Kids,
No Pets
Complete the
math madness
book that goes
with the story
you read.
Now, make a
math
madness
book based
on your
story about
kids and pets
or money
that comes
and goes.
Directions
are at the
author center
The Blue Contract
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
Read
Apply
Extend
Dinosaur
Before
Dark or
Airport
Control
Research a kind of
dinosaur or
airplane. Figure
out how big it is.
Graph its size on
graph paper or on
the blacktop
outside our room.
Label it by name
and size
Make a book
in which you
combine math
and dinosaurs
or airplanes,
or something
else big. It
can be a
number fact
book, a
counting
book, or a
problem
book.
Instructions
are at the
author center
Proportional Reasoning
Think-Tac-Toe
□
Create a word problem that
requires proportional
reasoning. Solve the
problem and explain why it
requires proportional
reasoning.
□
Find a word problem from
the text that requires
proportional reasoning.
Solve the problem and
explain why it was
proportional.
□
Think of a way that you use
proportional reasoning in your
life. Describe the situation,
explain why it is proportional
and how you use it.
□
Create a story about a
proportion in the world.
You can write it, act it,
video tape it, or another
story form.
□
How do you recognize a
proportional situation?
Find a way to think about
and explain proportionality.
□
Make a list of all the
proportional situations in the
world today.
□
Create a pict-o-gram, poem
or anagram of how to solve
proportional problems
□
Write a list of steps for
solving any proportional
problem.
□
Write a list of questions to ask
yourself, from encountering a
problem that may be
proportional through solving
it.
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
Similar Figures Menu
Imperatives (Do all 3):
1. 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.
2. Generate a list of applications for similar figures, and
similarity in general. Be sure to think beyond “find a
missing side…”
3. Develop a lesson to teach third grade students who are
just beginning to think about similarity.
Similar Figures Menu
Negotiables (Choose 1):
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.
2. 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:
1. 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.
2. Make a photo album showing the use of similar figures
in the world around us. Use captions to explain the
similarity in each picture.
3. Write a story about similar figures in a world without
similarity.
4. Write a song about the beauty and mathematics of
similar figures.
5. Create a “how-to” or book about finding and creating
similar figures.
Whatever it Takes!