differentiating mathematics in the elementary classroom raising student achievement conference st....
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Differentiating Mathematics in the Elementary Classroom
Raising Student Achievement ConferenceSt. 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 SmithEducational ConsultantCurriculum and Professional DevelopmentCave Creek, [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
Growth
InterestLearning Profile
Motivation Efficiency
Key Principles of a Differentiated Classroom
Key Principles of a Differentiated Classroom
• The teacher understands, appreciates, and builds upon student differences.
• 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 levelVaried supplementary materialsVaried scaffolding• reading• writing• research• technology
Tiered tasks and procedures Flexible time useSmall group instructionHomework optionsTiered or scaffolded assemssmentCompactingMentorshipsNegotiated criteria for qualityVaried graphic organizers
Providing support needed for a student to succeed in work slightly beyond his/her comfort zone.For example…
•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
1. Identify the learning objectives or standards ALL students must learn.
2. 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.
3. Plan and offer meaningful curriculum extensions for kids who qualify. **Depth and Complexity
Applications of the skill being taughtLearning Profile tasks based on understanding
the process instead of skill practiceDiffering perspectives, ideas across time,
thinking like a mathematician **Orbitals and Independent studies.
4. Eliminate all drill, practice, review, or preparation for students who have already mastered such things.
5. Keep accurate records of students’ compacting activities: document mastery.
Compacting
Strategy: Compacting
Developing a Tiered Activity
Select the activity organizer•concept•generalization
Essential to buildinga framework ofunderstanding
Think about your students/use assessments
• readiness range• interests• learning profile• talents
skillsreadingthinkinginformation
Create an activity that is• interesting• high level• causes students to use key skill(s) to understand a key idea
Chart the complexity of the activity
High skill/Complexity
Low skill/complexity
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
Match task to student based on student profile and task requirements
1
3
5
2
4
6
Information, Ideas, Materials, Applications
Representations, Ideas, Applications, Materials
Resources, Research, Issues, Problems, Skills, Goals
Directions, Problems, Application, Solutions, Approaches, Disciplinary Connections
Application, Insight, Transfer
Solutions, Decisions, Approaches
Planning, Designing, Monitoring
Pace of Study, Pace of Thought
The Equalizer
1. Foundational Transformational
2. Concrete Abstract
1. Simple Complex
2. Single Facet Multiple Facets
3. Small Leap Great Leap
4. More Structured More Open
5. Less Independence Greater Independence
6. Slow Quick
Adding FractionsGreen 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.
Blue GroupManipulatives 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.
Red GroupUse 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.
BRAIN RESEARCH SHOWS THAT. . .Eric Jensen, Teaching With the Brain in Mind, 1998
Choices vs. Required content, process, product no student voice
groups, resources environment restricted resources
Relevant vs. Irrelevant meaningful impersonal
connected to learner out of context deep understanding only to pass a test
Engaging vs. Passive emotional, energetic low interaction
hands on, learner input lecture seatwork
EQUALSIncreased intrinsic Increased MOTIVATION 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. Yes No2. I am able to ignore the noise of
other people talking while I am working. Yes No3. I like to work at a table or desk. Yes No4. I like to work on the floor. Yes No5. I work hard by myself. Yes No6. I work hard for my parents or teacher. Yes No7. I will work on an assignment until it is completed, no
matter what. Yes No8. Sometimes I get frustrated with my work
and do not finish it. Yes No9. When my teacher gives an assignment, I like to
have exact steps on how to complete it. Yes No10. When my teacher gives an assignment, I like to
create my own steps on how to complete it. Yes No11. I like to work by myself. Yes No12. I like to work in pairs or in groups. Yes No13. I like to have unlimited amount of time to work on
an assignment. Yes No14. I like to have a certain amount of time to work on
an assignment. Yes No15. I like to learn by moving and doing. Yes No16. I like to learn while sitting at my desk. 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 multi-media 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
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. ___
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
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
Instructions: My Way …A Profile
Write your score beside each number. Add each Row to determine your expression style profile.
Learner Profile Card
Auditory, Visual, Kinesthetic
Modality
Multiple Intelligence Preference
Gardner
Analytical, Creative, Practical
Sternberg
Student’s Interests
Array Inventory
Gender Stripe
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 orientationgroup/peer orientation
adult orientationcombination
Learning Environment
quiet/noisewarm/coolstill/mobile
flexible/fixed“busy”/”spare”
Cognitive Style
Creative/conformingEssence/facts
Expressive/controlledNonlinear/linear
Inductive/deductivePeople-oriented/task or Object oriented
Concrete/abstractCollaboration/competitionInterpersonal/introspective
Easily distracted/long Attention spanGroup achievement/personal achievement
Oral/visual/kinestheticReflective/action-oriented
Intelligence Preference
analyticpracticalcreative
verbal/linguisticlogical/mathematical
spatial/visualbodily/kinestheticmusical/rhythmic
interpersonalintrapersonal
naturalistexistential
Gender &Culture
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 PreferencesDirections: 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.
12 3
45
67
8
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.
Smith & Smarr, 2005
12 3
456
78
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)
Smith & Smarr, 2005
12 3
45
67
8
EIGHT STYLES OF LEARNINGTYPE CHARACTERISTICS LIKES TO IS GOOD AT LEARNS BEST BY
LINGUISTIC
LEARNER“The Word Player”
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
LOGICAL/
Mathematical
Learner“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“The Visualizer”
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
MUSICAL LEARNER“The Music Lover”
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
EIGHT STYLES OF LEARNING, Cont’d
TYPE CHARACTERISTICS LIKES TO IS GOOD AT LEARNS BEST BY
BODILY/
Kinesthetic
Learner“The Mover”
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
INTERpersonal
Learner“The Socializer”
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
INTRApersonal
Learner“The Individual”
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
NATURALIST“The Nature Lover”
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
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.
Linear – Schoolhouse Smart - SequentialANALYTICALThinking About the Sternberg Intelligences
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 _________________.
Streetsmart – Contextual – Focus on UsePRACTICAL
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 IntelligencesRobert Sternberg
Mark each sentence T if you like to do the activity and F if you do not like to do the activity.
1. Analyzing characters when I’m reading or listening to a story ___2. Designing new things ___
3. Taking things apart and fixing them ___4. Comparing and contrasting points of view ___5. Coming up with ideas ___6. Learning through hands-on activities ___7. Criticizing my own and other kids’ work ___8. Using my imagination ___9. Putting into practice things I learned ___10. Thinking clearly and analytically ___11. Thinking of alternative solutions ___12. Working with people in teams or groups ___13. Solving logical problems ___14. Noticing things others often ignore ___15. Resolving conflicts ___
Triarchic Theory of IntelligencesRobert Sternberg
Mark each sentence T if you like to do the activity and F if you do not like to do the activity.
16. Evaluating my own and other’s points of view ___17. Thinking in pictures and images ___18. Advising friends on their problems ___19. Explaining difficult ideas or problems to others ___20. Supposing things were different ___21. Convincing someone to do something ___22. Making inferences and deriving conclusions ___23. Drawing ___24. Learning by interacting with others ___25. Sorting and classifying ___26. Inventing new words, games, approaches ___27. Applying my knowledge ___28. Using graphic organizers or images to organize your thoughts ___29. Composing ___30. Adapting to new situations ___
Triarchic Theory of Intelligences – KeyRobert Sternberg
Transfer your answers from the survey to the key. The column with the most True responses is your dominant intelligence.
Analytical Creative Practical1. ___ 2. ___ 3. ___4. ___ 5. ___ 6. ___7. ___ 8. ___ 9. ___10. ___ 11. ___ 12. ___13. ___ 14. ___ 15. ___16. ___ 17. ___ 18. ___19. ___ 20. ___ 21. ___22. ___ 23. ___ 24. ___25. ___ 26. ___ 27. ___28. ___ 29. ___ 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
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.
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.
Understanding Order of Operations
Analytic Task
Practical Task
Creative Task
Make a chart that shows all ways you can think of to use order of operations to equal 18.
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.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
Key Principles of a Differentiated Classroom
• AssessmentAssessment and and instructioninstruction are are inseparableinseparable..
• AssessmentAssessment and and instructioninstruction are are inseparableinseparable..
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 SOURCES1. Journal entry2. Short answer test3. Open response test4. Home learning5. Notebook6. Oral response7. Portfolio entry8. Exhibition9. Culminating product10. Question writing11. Problem solving
TEACHER DATA MECHANISMS
1. Anecdotal records2. Observation by checklist3. Skills checklist4. Class discussion5. Small group interaction6. Teacher – student
conference7. Assessment stations8. Exit cards9. Problem posing10. Performance tasks and
rubrics
Key Principles of a Differentiated Classroom
Key Principles of a Differentiated Classroom
• The teacher adjusts The teacher adjusts content, content, process, and productprocess, and product in response to in response to student student readiness, interestsreadiness, interests, and , and learning profilelearning profile..
• The teacher adjusts The teacher adjusts content, content, process, and productprocess, and product in response to in response to student student readiness, interestsreadiness, interests, and , and learning profilelearning 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.• Add distractors to outside edges and
“letter” pieces at the end.
m=3
b=6 -2/3
Nanci Smith
The ROLE of writer, speaker,artist, historian, etc.
An AUDIENCE of fellow writers,students, citizens, characters, etc.
Through a FORMAT that is written, spoken, drawn, acted, etc.
A TOPIC related to curriculumcontent in greater depth.
electron
neutron
proton
R A F T
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 RAFTRole 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 Supplementaryangle
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 Keep me in mind when solving an
equation
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!
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
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.
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.
CubingCubing
Cubing
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.
Nanci Smith
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.
5
3
5
1
2
1
3
1
15
11
5
2
3
1
Nanci Smith
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.
91
1
7
3
13
2
7
1
7
3 and
2
1
3
1
Designing a Differentiated Learning Designing a Differentiated Learning ContractContract
A Learning Contract has the following components1.1. A Skills ComponentA Skills Component
Focus is on skills-based tasksAssignments are based on pre-assessment of students’ readinessStudents work at their own level and pace
2.2. A content componentA content componentFocus is on applying, extending, or enriching key content (ideas, understandings)Requires sense making and productionAssignment is based on readiness or interest
3.3. A Time LineA Time LineTeacher sets completion date and check-in requirementsStudents select order of work (except for required meetings and homework)
4. The AgreementThe AgreementThe teacher agrees to let students have freedom to plan their timeStudents agree to use the time responsiblyGuidelines for working are spelled outConsequences for ineffective use of freedom are delineatedSignatures of the teacher, student and parent (if appropriate) are placed on the agreement
Differentiating Instruction: Facilitator’s Guide, ASCD, 1997
Personal AgendaPersonal Agenda for _______________________________________
Starting Date _____________________________________________________
Teacher & studentinitials at completion
TaskSpecial Instructions
Remember to complete your daily planning log; I’ll call on you for conferences & instructions.
Montgomery County, MD
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.
Nanci Smith
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.
5
3
5
1
2
1
3
1
15
11
5
2
3
1
Nanci Smith
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.
91
1
7
3
13
2
7
1
7
3 and
2
1
3
1
Level 1:1. a, b, c and d each represent a different value. If a = 2, find b, c, and d.
a + b = ca – c = da + 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 = cb + b = dc – 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 = bb - a = ccd = -dd + 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
. 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.
7513 xx
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.
Learning Contracts
Strategy: Learning Contracts
Designing a Differentiated Learning Designing a Differentiated Learning ContractContract
A Learning Contract has the following components1.1. A Skills ComponentA Skills Component
Focus is on skills-based tasksAssignments are based on pre-assessment of students’ readinessStudents work at their own level and pace
2.2. A content componentA content componentFocus is on applying, extending, or enriching key content (ideas, understandings)Requires sense making and productionAssignment is based on readiness or interest
3.3. A Time LineA Time LineTeacher sets completion date and check-in requirementsStudents select order of work (except for required meetings and homework)
4. The AgreementThe AgreementThe teacher agrees to let students have freedom to plan their timeStudents agree to use the time responsiblyGuidelines for working are spelled outConsequences for ineffective use of freedom are delineatedSignatures of the teacher, student and parent (if appropriate) are placed on the agreement
Differentiating Instruction: Facilitator’s Guide, ASCD, 1997
Personal AgendaPersonal Agenda for _______________________________________
Starting Date _____________________________________________________
Teacher & studentinitials at completion
TaskSpecial Instructions
Remember to complete your daily planning log; I’ll call on you for conferences & instructions.
Montgomery County, MD
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 ContractKey Skills: Graphing and MeasuringKey Concepts: Relative SizesNote 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
measurement and at least one graph. Turn it into
a book at the author center
Solve the
great graph
mystery in your
math folder. Check
Your answers with a
buddy, then with the teacher
Work at the measuring
and graphing center until
you complete the red
workDesign an
animal on graph
paper using the
creature blueprint.
Get your graph
approved. Then make a
drawing, painting, or
model of it
Use the dominoes to
solve the problems in
your folder. Draw
and then write
your answers
Come to the red
math workshop
on Monday
and
Tuesday
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
The Green ContractKey Skills: Graphing and MeasuringKey Concepts: Relative SizesNote to User: This is a Grade 3 math contract for students at or near grade level in these skills
Read Apply Extend
Alexander 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
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
Work at the measuring
and graphing center until
you complete the green
workDesign an
animal on graph
paper using the
creature blueprint.
Get your graph
approved. Then make a
drawing, painting, or
model of it
Complete the dominoes
multiplication challenge.
Record your answers
on the wall chart
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 yourwork
The Blue ContractKey Skills: Graphing and MeasuringKey Concepts: Relative SizesNote 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
Solve the
graph mystery
in your folder. You
can work with someone
on the blue team if you’d
like.
Work at the measuring
and graphing center until
you complete the blue
workFind aplace in
our school to
make a pattern
graph of. Make the
graph and create three
problems for a
classmate
to solve.
Do a timed test of
Two-digit multiplication. Use a
peer monitor
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 peermonitor to
audit yourwork.
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!