Differentiating Mathematics at the Middle and High School Levels Raising Student Achievement Conference St. Charles, IL

1. Differentiating Mathematics at the Middle and High School LevelsRaising Student Achievement ConferenceSt. Charles, ILDecember 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 nanci_mathmaster@yahoo.com

2. Differentiation of Instruction Is a teacher’s response to learner’s needsguided by general principles of differentiation Respectful tasks Flexible grouping Continual assessment Teachers Can Differentiate Through: Process Product Content According to Students’ Readiness Interest Learning Profile

3. What’s the point of differentiating in these different ways? Learning Profile Readiness Interest Growth Motivation Efficiency

4. 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

5. READINESS What does READINESS mean? It is the student’s entry point relative to a particular understanding or skill. C.A.Tomlinson, 1999

6. A Few Routes toREADINESS 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

7. Providing support needed for a student to succeed in work slightly beyond his/her comfort zone. Scaffolding • 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

8. Compacting • 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

9. Create an activity that is • interesting • high level • causes students to use • key skill(s) to understand • a key idea High skill/ Complexity Low skill/ complexity Chart the complexity of the activity • 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 Developing a Tiered Activity 1 2 • Select the activity organizer • concept • generalization • Think about your students/use assessments • readiness range • interests • learning profile • talents Essential to building a framework of understanding skills reading thinking information 3 4 5 6

10. The Equalizer • Foundational Transformational • Concrete Abstract • Simple Complex • Single Facet Multiple Facets • Small Leap Great Leap • More Structured More Open • Less Independence Greater Independence • Slow Quick 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

11. 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. 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. Adding Fractions 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.

12. Graphing with a Point and a Slope All groups: • Given three equations in slope-intercept form, the students will graph the lines using a T-chart. Then they will answer the following questions: • What is the slope of the line? • Where is slope found in the equation? • Where does the line cross the y-axis? • What is the y-value of the point when x=0? (This is the y-intercept.) • Where is the y-value found in the equation? • Why do you think this form of the equation is called the “slope-intercept?”

13. Graphing with a Point and a Slope Struggling Learners: Given the points • (-2,-3), (1,1), and (3,5), the students will plot the points and sketch the line. Then they will answer the following questions: • What is the slope of the line? • Where does the line cross the y-axis? • Write the equation of the line. The students working on this particular task should repeat this process given two or three more points and/or a point and a slope. They will then create an explanation for how to graph a line starting with the equation and without finding any points using a T-chart.

14. Graphing with a Point and a Slope Grade-Level Learners: Given an equation of a line in slope-intercept form (or several equations), the students in this group will: • Identify the slope in the equation. • Identify the y-intercept in the equation. • Write the y-intercept in coordinate form (0,y) and plot the point on the y-axis. • use slope to find two additional points that will be on the line. • Sketch the line. When the students have completed the above tasks, they will summarize a way to graph a line from an equation without using a T-chart.

15. Graphing with a Point and a Slope Advanced Learners: Given the slope-intercept form of the equation of a line, y=mx+b, the students will answer the following questions: • The slope of the line is represented by which variable? • The y-intercept is the point where the graph crosses the y-axis. What is the x-coordinate of the y-intercept? Why will this always be true? • The y-coordinate of the y-intercept is represented by which variable in the slope-intercept form? Next, the students in this group will complete the following tasks given equations in slope-intercept form: • Identify the slope and the y-intercept. • Plot the y-intercept. • Use the slope to count rise and run in order to find the second and third points. • Graph the line.

16. 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 EQUALS Increased intrinsic Increased MOTIVATION APATHY & RESENTMENT

17. -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!!

18. 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.”

19. How Do You Like to Learn? 1. I study best when it is quiet. Yes No 2. I am able to ignore the noise of other people talking while I am working.Yes No 3. I like to work at a table or desk. Yes No 4. I like to work on the floor. Yes No 5. I work hard by myself. Yes No 6. I work hard for my parents or teacher. Yes No 7. I will work on an assignment until it is completed, no matter what. Yes No 8. Sometimes I get frustrated with my work and do not finish it. Yes No 9. When my teacher gives an assignment, I like to have exact steps on how to complete it. Yes No 10. When my teacher gives an assignment, I like to create my own steps on how to complete it. Yes No 11. I like to work by myself. Yes No 12. I like to work in pairs or in groups. Yes No 13. I like to have unlimited amount of time to work on an assignment. Yes No 14. I like to have a certain amount of time to work on an assignment. Yes No 15. I like to learn by moving and doing. Yes No 16. I like to learn while sitting at my desk. Yes No

20. 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).

23. Instructions: My Way …A Profile Write your score beside each number. Add each Row to determine your expression style profile.

24. Learner Profile Card Gender Stripe Auditory, Visual, Kinesthetic Modality Analytical, Creative, Practical Sternberg Student’s Interests Multiple Intelligence Preference Gardner Array Inventory Nanci Smith,Scottsdale,AZ

25. 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.

26. Learning Profile Factors Learning Environment quiet/noise warm/cool still/mobile flexible/fixed “busy”/”spare” Group Orientation independent/self orientation group/peer orientation adult orientation combination Gender & Culture Intelligence Preference analytic practical creative verbal/linguistic logical/mathematical spatial/visual bodily/kinesthetic musical/rhythmic interpersonal intrapersonal naturalist existential 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

27. 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. • I Prefer reading a story rather than listening to someone tell it. A D • I would rather watch television than listen to the radio. A D • I remember faces better than names. A D • I like classrooms with lots of posters and pictures around the room. A D • The appearance of my handwriting is important to me. A D • I think more often in pictures. A D • I am distracted by visual disorder or movement. A D • I have difficulty remembering directions that were told to me. A D • I would rather watch athletic events than participate in them. A D • I tend to organize my thoughts by writing them down. A D • My facial expression is a good indicator of my emotions. A D • I tend to remember names better than faces. A D • I would enjoy taking part in dramatic events like plays. A D • I tend to sub vocalize and think in sounds. A D • I am easily distracted by sounds. A D • I easily forget what I read unless I talk about it. A D • I would rather listen to the radio than watch TV A D • My handwriting is not very good. A D • When faced with a problem , I tend to talk it through. A D • I express my emotions verbally. A D • I would rather be in a group discussion than read about a topic. A D

28. I prefer talking on the phone rather than writing a letter to someone. A D • I would rather participate in athletic events than watch them. A D • I prefer going to museums where I can touch the exhibits. A D • My handwriting deteriorates when the space becomes smaller. A D • My mental pictures are usually accompanied by movement. A D • I like being outdoors and doing things like biking, camping, swimming, hiking etc. A D • I remember best what was done rather then what was seen or talked about. A D • When faced with a problem, I often select the solution involving the greatest activity. A D • I like to make models or other hand crafted items. A D • I would rather do experiments rather then read about them. A D • My body language is a good indicator of my emotions. A D • 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.

29. 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. 1 2 3 4 5 6 8 7 Smith & Smarr, 2005

30. 1 3 2 4 5 8 6 7 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

31. 1 3 2 4 5 8 6 7 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

32. EIGHT STYLES OF LEARNING

33. EIGHT STYLES OF LEARNING, Cont’d

34. Introduction to Change(MI) • Logical/Mathematical Learners: Given a set of data that changes, such as population for your city or town over time, decide on several ways to present the information. Make a chart that shows the various ways you can present the information to the class. Discuss as a group which representation you think is most effective. Why is it most effective? Is the change you are representing constant or variable? Which representation best shows this? Be ready to share your ideas with the class.

35. Introduction to Change(MI) • Interpersonal Learners:Brainstorm things that change constantly. Generate a list. Discuss which of the things change quickly and which of them change slowly. What would graphs of your ideas look like? Be ready to share your ideas with the class.

36. Introduction to Change(MI) • Visual/Spatial Learners: Given a variety of graphs, discuss what changes each one is representing. Are the changes constant or variable? How can you tell? Hypothesize how graphs showing constant and variable changes differ from one another. Be ready to share your ideas with the class.

37. Introduction to Change(MI) • Verbal/Linguistic Learners: Examine articles from newspapers or magazines about a situation that involves change and discuss what is changing. What is this change occurring in relation to? For example, is this change related to time, money, etc.? What kind of change is it: constant or variable? Write a summary paragraph that discusses the change and share it with the class.

38. Multiple Intelligence Ideas for Proofs! • Logical Mathematical: Generate proofs for given theorems. Be ready to explain! • Verbal Linguistic: Write in paragraph form why the theorems are true. Explain what we need to think about before using the theorem. • Visual Spatial: Use pictures to explain the theorem.

39. Multiple Intelligence Ideas for Proofs! • Musical: Create a jingle or rap to sing the theorems! • Kinesthetic: Use Geometer Sketchpad or other computer software to discover the theorems. • Intrapersonal: Write a journal entry for yourself explaining why the theorem is true, how they make sense, and a tip for remembering them.

40. 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.

41. 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 _________________. Streetsmart – Contextual – Focus on Use PRACTICAL 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 ____________.

42. 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. • 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 ___

43. 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. • 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 ___ 30. Adapting to new situations ___

44. 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 Practical 1. ___ 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 _____

45. Understanding Order of Operations Make a chart that shows all ways you can think of to use order of operations to equal 18. Analytic 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. Practical Task Creative Task Write a book of riddles that involve order of operations. Show the solution and pictures on the page that follows each riddle.

46. Forms of Equations of Lines • Analytical Intelligence: Compare and contrast the various forms of equations of lines. Create a flow chart, a table, or any other product to present your ideas to the class. Be sure to consider the advantages and disadvantages of each form. • Practical Intelligence: Decide how and when each form of the equation of a line should be used. When is it best to use which? What are the strengths and weaknesses of each form? Find a way to present your conclusions to the class. • Creative Intelligence: Put each form of the equation of a line on trial. Prosecutors should try to convince the jury that a form is not needed, while the defense should defend its usefulness. Enact your trial with group members playing the various forms of the equations, the prosecuting attorneys, and the defense attorneys. The rest of the class will be the jury, and the teacher will be the judge.

47. Circle Vocabulary All Students: Students find definitions for a list of vocabulary (center, radius, chord, secant, diameter, tangent point of tangency, congruent circles, concentric circles, inscribed and circumscribed circles). They can use textbooks, internet, dictionaries or any other source to find their definitions.

48. Circle Vocabulary Analytical Students make a poster to explain the definitions in their own words. Posters should include diagrams, and be easily understood by a student in the fifth grade. Practical Students find examples of each definition in the room, looking out the window, or thinking about where in the world you would see each term. They can make a mural, picture book, travel brochure, or any other idea to show where in the world these terms can be seen.

49. Circle Vocabulary Creative Find a way to help us remember all this vocabulary! You can create a skit by becoming each term, and talking about who you are and how you relate to each other, draw pictures, make a collage, or any other way of which you can think. OR Role Audience Format Topic Diameter Radius email Twice as nice Circle Tangent poem You touch me! Secant Chord voicemail I extend you.

50. Key Principles of a Differentiated Classroom • Assessment and instruction are inseparable. Source: Tomlinson, C. (2000). Differentiating Instruction for Academic Diversity. San Antonio, TX: ASCD