2. Reach Every Student Through Differentiation
Great teachers of Mathematics help their students love Math! They differentiate to provide appropriate learning experiences for all their learners: A talented student might be on a learning contract investigating more in depth understandings of the topic, while another learner will work fewer problems with more concrete support, all along the way. Tiered assignments, differentiated homework, compacting, learning stations, small groups, choice boards, Math journals may all be implemented over a period of time in Math class.
In our country, we excel at creative thinking and application to real world problems. We have produced many great minds and great inventors.
As the advent of technology increases the pace of change in our world, we must begin to develop “tomorrow’s minds” today. Those minds must consist of a strong base of mathematical understanding. As educators we must excel in teaching tomorrow’s minds, today…
Through differentiation.
Comes from differentiating instruction day 3 session 2 presented by (FIN) The Florida Inclusion Network and (FDLRS) The Florida Diagnostic and Learning Resources System

3. Corners Differentiated Activity
Go to the corner that best describes your learning Style and share with the participants in your corner:
Prior experience with Differentiated Instruction
An example of how your learner preferences affect your learning and teaching
Grouped by learner preferences for this activity

4. What is your Mathematical Learning Style?
The Mastery Style: People in this category tend to work step–by–step.
The Understanding Style: People in this category tend to search for patterns, categories, and reasons.
The Interpersonal Style: People in this category tend to learn through conversation and personal relationship and association.
The Self-Expressive Style: People in this category tend to visualize and create images and pursue multiple strategies.
Differentiating Mathematics Instruction
The purpose of differentiating instruction in all subject areas is to engage students in instruction and learning in the classroom. All students need sufficient time and a variety of problem-solving contexts to use concepts, procedures and strategies and to develop and consolidate their understanding. When teachers are aware of their students’ prior knowledge and experiences, they can consider the different ways that students learn without pre-defining their capacity for learning. Differentiation is not a strategy but rather a way of teaching that accommodates differences among children so that all are learning on an ongoing basis.

5. Mathematical Learning
Students who favor the Mastery style learn most easily from teaching approaches that emphasize step-by-step demonstrations and repetitive practices. They struggle with abstractions, explanations, and non-routine problem solving.
Students who favor the Understanding style learn most easily from teaching approaches that emphasize concepts and the reasoning behind mathematical operations. These students struggle with work that emphasizes collaboration, application, and routine drill and practice.

6. Mathematical Learning
Students who favor the Interpersonal style learn most easily from teaching approaches that emphasize cooperative learning, real-life contexts, and connections to everyday life. This group struggles with independent seatwork, abstraction, and out-of-context, non-routine problem solving.
Students who favor the Self-Expressive style learn most easily from teaching approaches that emphasize visualization and exploration. These students struggle with step-by-step computation and routine drill and practice.
Educational Leadership feb.2004 vol.61,pgs
In general, we tend to develop preferences or strengths in one or two of the styles and develop weaknesses in the ones that remain.

7. What is Differentiation?
“At its most basic level, differentiating instruction means “shaking up” what goes on in the classroom so that students have multiple options for taking in information, making sense of ideas and expressing what they learn… a differentiated classroom provides different avenues to acquiring content, to processing and making sense of ideas, and to developing products so that each student can learn effectively.” (Carol Tomlinson, 1999, p.1)
Differentiating in Mathematics
• The focus of instruction must be on the key mathematics concepts (big ideas) being taught. • There must be some aspect of choice for
the student, in terms of the details of the learning task, the ways the task can be carried out and how the task is assessed
• Assessment for learning is essential to determine the learning needs of different students.
(e.g., Dacey & Lynch, 2007; Dacey & Salemi, 2007; Small, in press b)

8. Today’s Student Diverse Students
These are just some of the differences in students that are found in Florida’s classrooms. There are others, but this makes a point: that one size may not fit All!!

9. Reasons for DI Gap in achievement levels
Focused and coherent curriculum
Increased student expectations
Higher demand for mathematical skills
Math solutions

10. Teachers differentiate by:
WHAT we want students to learn and HOW we give them access to it.
HOW a student makes sense of the learning.
Add notes on Content – process – product see behind slides
WHAT a student makes or does that SHOWS he/she has the knowledge,
understanding, and skills that were taught.

12. Know your Students Learner Profiles Student learning styles Interests
Interviews
Questionnaires
Readiness
Formal/informal assessments
Anecdotal records
Use diagnostic assessment to determine student readiness. Can be formal or informal. First step in implementation
Determine student interest. Teachers can then try to incorporate these interests into their lessons.
Identify student learning styles.

13. Create a Responsive Learning Environment
Know your learners
More than desk alignment
Provide a safe place to make mistakes and learn from them
Meet your learners “where they are”
Think about how you can Make Math Irresistible!
In a math class this may mean designing a physical environment that as conducive to learning. Developing routines and procedures that ensure smooth daily transitions from one place to another in the room.

14. Learning Goals
Analogy about driving to a new place without a map – the goal is to end up in the same location but may take different routes to get their – some short cuts – some with longer destinations and assistance.
Designing and aligning student experiences to maximize Mathematical understanding
Developing the understanding that all of the diverse learners in our group will be moving forward together, sometimes all together, sometimes in different ways, but all heading to mastery of this particular content. Etc.
Need to make animation

15. respectful tasks ongoing assessment & adjustment flexible groups
DIFFERENTIATION…
is the proactive acceptance of and planning for student differences, including their
readiness interests learning profiles
Teachers can respond to student differences by differentiating
content process products environment
while always keeping in mind the guiding principles of
respectful tasks ongoing assessment & adjustment flexible groups

16. Differentiating Instruction C - R- A
“… allows all students to access the same classroom curriculum by providing entry points, learning tasks, and outcomes that are tailored to the students’ needs.”
(Hall, Strangman, & Meyer, 2003)
Just a brief overview of how teachers should “teach” and how students “learn Mathematics”. How does this statement reflect your current practices…
Concrete
Representational
Abstract

17. “I actually think that the most important thing that teachers should be thinking about is not the activity that they choose, it’s the questions they ask.”
Marian Small is Dean Emerita of the University of New Brunswick in Canada
“Love Learning with Dr. Marian Small” video 5:51 minutes

18. Developing Mathematical Thinking with Effective Questions
To help students build confidence and rely on their own understanding, ask…
To help students learn to reason
mathematically, ask…
To check student progress, ask…
To help students collectively make sense of mathematics, ask…
To encourage conjecturing, ask…
To promote problem solving, ask…
To help when students get stuck, ask…
To make connections among ideas and applications, ask…
To encourage reflection, ask…
TeacherLine
PBS.org/teacherline
Funded by a grant from the U.S. DOE

19. “Beyond One Right Answer” Dr. Marian Small
“Differentiating instruction is a great way to make math meaningful for all. It's just a question of the questions teachers pose.”
Dr. Marian Small is an author and International Professional Development Consultant.
“Leading Guru of differentiating instruction in mathematics.”
Hyperlink to article, if needed.

20. Consider the following two scenarios. Do they sound familiar?
A teacher decides that she wants a math lesson to focus on two-digit by two-digit multiplication.
She finds an appropriate problem for the students to work on. Although she knows that six or seven students still struggle with the concepts involved in multiplying by even a single-digit number, she presents the problem to all students, making sure that the struggling students receive help from herself or other students.
A teacher is working on teaching fact families.
He asks students to describe the fact family for One student offers a response: , 4 + 3, 7 - 4, The teacher records this on the board and checks that other students concur. The whole episode takes less than five minutes, only one student responded, and now the teacher needs to set up another activity.
From article, “Beyond One Right Answer” by Marian Small
COMMENTS about Scenario 1:
To her credit, the teacher is attempting to provide support. However, she's not only putting students in a situation that might reinforce their belief that math is just too hard for them, but she's also implying that she doesn't expect them to succeed without help. Is that the message she really wants to deliver?
COMMENTS about Scenario 2:
The teacher is focusing on an important mathematical idea—the relationship between addition and subtraction— but in a fairly narrow way. Students come to view math as a subject in which their job is to quickly answer question after question with the single right answer the teacher expects.

21. Two Beliefs That Need to Change
…all students should work on the same problem at the same time (Scenario 1)
each math question should have a single answer (Scenario 2)
Both scenarios
reflect common practice among many hardworking and capable teachers. But what else can teachers really do?
COMMENTS ABOUT THE TITLE OF THE SLIDE… read this before clicking to show the other information on the slide.
Teaching mathematics at the elementary and middle school levels has changed in many ways in the last two decades. Students are more likely to use manipulatives and technology than in the past, teachers are more likely to encourage students to use personal strategies, and there is typically much more discussion in the classroom.
CLICK TO SHOW THE BELIEFS.
Then CLICK TO SHOW THE NEXT PART.
READ THIS FROM SLIDE, “Both scenarios reflect common practice among many hardworking and capable teachers. But what else can teachers really do?”

22. An Idea Takes Root
Background-
Two “universes collided” whenever the research of one of Dr. Small’s graduate students (the kinds of questions that math teachers ask during instruction) and what she was working on at that time (researching the various phases of student development in each strand of mathematics). Dr. Small noticed that there was potential in using questioning as a way to differentiate instruction in a classroom with groups of students at different levels.
What emerged was the delineation of two core techniques for differentiating instruction in mathematics in a meaningful, but manageable way.
From article, “Beyond One Right Answer” by Marian Small
“Many teachers shy away from differentiation in math because they do not see how to do it.
These two strategies—creating open questions and creating parallel tasks—show how to differentiate math instruction in a manageable way. By doing so, teachers can make all students feel like part of the larger community of learners as all contribute to a rich discussion of mathematics.”

23. Open-Ended Questions
Teachers create open questions by allowing for a certain level of ambiguity.
Students may initially be a little uncomfortable with ambiguity, but they almost always ‘warm up to’ and appreciate the latitude that the ambiguity allows.
Information from article, “Beyond One Right Answer”. (Saved in folder pdf documents for PPT Summer2014).
COMMENTS ABOUT THE SLIDE INFO…
Open questions also provide choice, one of the elements implicit in differentiating instruction. Students can answer in a way that is suitable for their level. And everyone benefits from different perspectives when they hear other students respond. The open question is both accessible to and enriching for all.
Some teachers may worry that students will not sufficiently challenge themselves—for example, providing simple responses when they are capable of more complex ones. In practice, this happens much less often than one might think; students seem to enjoy challenging themselves when they have the latitude to do so.
Examples of creating open questions…
Rather than asking for two numbers that add up to 37, a teacher could ask for two numbers that add up to about 40. Or instead of asking for the third angle size in a triangle with one angle of 20° and another of 38°, a teacher could ask for three possible angle sizes in a triangle with at least one narrow (or sharp) angle.
Strategy 1: Start with the answer. A teacher can take a straightforward question and present it backward. For example, instead of asking,
"What is ?" a teacher could say, "I added two numbers. The sum is 61. What numbers might I have added?“
The area of a rectangle is 20 square inches. What might be its length and width?
A 3D shape has 8 vertices. What might it look like? (Students might suggest a cube.)
Strategy 2: Ask for similarities and differences. Asking students how two things are alike and how they are different can provide teachers with valuable assessment for learning information.
How are the numbers 4 and 9 (or 350 and 550 or 100 and 1,000, and so on) alike? How are they different? (Students might point out whether the numbers are even or odd or divisible by numbers other than themselves.)
How is the formula for the perimeter of a rectangle like the formula for its area? How is it different? (Students might indicate that both formulas involve using values for the length and width of the rectangle, but that one involves addition and the other doesn't.)

24. Parallel Tasks Focusing on the same “big ideas”, but different levels of difficulty
Strategy 1:
Strategy 2:
Teacher may let students choose between two problems which have been written with different levels of difficulty.
Pose common questions for all students to answer. The teacher could ask questions, no matter which task the student(s) completed.
The questions focus on common elements; however, asking students to describe their specific strategies gives opportunities to differentiate.
Focusing on the same “big ideas”, but different levels of difficulty…
For example, using multiplication to simplify the counting of equal groups is useful no matter the size of the numbers involved in the computation. Parallel tasks might allow students who are ready to deal with only simpler values to use those simpler values, whereas students who are ready for more complex work could use more challenging values. Common questions that focus on strategies and the meaning of multiplication would apply to both tasks.
Strategy 1:
 Choice 1: There are 427 students in Tara's school in the morning. Ninety-nine of them left for a field trip. How many students are still in their classrooms? (The problem involves subtraction and is suitable for mental math calculations because 99 is so close to 100.)
 Choice 2: There are 61 students in 3rd grade. Nineteen of them are in the library. How many students are still in the classrooms? (This problem also uses subtraction and is suitable for mental math, but it involves smaller values for students who are not ready for work with 3-digit numbers.)
Strategy 2:
 Before you calculated, could you tell whether the number of students left in the classrooms would be more or less than one-half of the total number of students? Explain.
 What operation did you or could you use to solve your problem? Why that one?
 Would it be easier to solve the problem if one more student had left the classroom? Why?
 How could you use mental math to solve your problem?
 How did you solve your problem? How many students are still in their classrooms?

25. Louisiana Believes Department of Education Video Library
7th Grade
High School
Summary:
In this 7th grade Math lesson, students categorize expressions, equations, and inequalities and discuss how they know how to categorize them (video 5:30 min.)
Video Notes/Reflection
Summary:
By the end of the class, to be able to simplify rational expressions (video 9:28 min.)
Video Notes/Reflection
Activity to Focus on Listening to Teacher Questioning and Student Engagement.
View 7th Grade Video, then pass out the Video Notes for that video and have teachers briefly discuss the documented information. Any comments they would like to make to the whole group?
View High School Video, then pass out the Video Notes for that video and have teachers briefly discuss the documented information. Any comments they’d like to make to the whole group?
7th Grade Video Notes from lesson on categorizing expressions
High School Video Notes from lesson on Rational Expressions

26.
Page 7of the CCSSM content standards
Needs to be underlined from line 7 The standards should…to line 10 education needs. Also line 14 No set of grade… to the end of the page.

27. Differentiation using Tiered Lessons
“Tiered activities are really quite essential. They are almost the meat and potatoes of differentiation.” (Tomlinson)
As noted from the previous slide from CCSSM one of the ways to address understanding is having students work on the same concept at different levels of complexity and with different levels of support or open-endedness. This does not mean more work or less work, just different work.

28. What is Tiered Instruction?
By keeping the focus of the
activity the same, but
providing routes of access at
varying degrees of difficulty,
the teacher maximizes the
likelihood that:
each student comes away
with pivotal skills &
understandings
2) each student is appropriately
challenged.
Teachers use tiered activities so that all students focus on
essential understandings and skills but at different levels of complexity, abstractness, and open-endedness.

30. When Tiering Instruction:
Adjust the…
Level of Complexity
Amount of Structure
Materials
Time/Place
Number of Steps
Form of Expression
Level of Dependence
Questions to guide your thinking as you plan for lessons. How can I ensure that each student experiences challenges? How can I scaffold learning to increase the likelihood of success? In what different ways can my students demonstrate their new understanding? Are there choices students can make??

31. Example of a tiered lesson K-2

32. Open-Ended Probes
How do you describe a cube to someone who has never seen one?
The answer is 87. What could the question be?
How is measurement used in your home?
How might we write numbers if we didn’t have zeroes?
Imagine you are trying to help someone understand what three-tenths means.
What pictures could you draw to be helpful?
How many different pictures can you make?

33. Strategies for
Strategies for differentiating middle school math july 13,2010 James and Tammy Pasons Nashville Public Schools

34. Subject: Mathematics - Statistics Grade: Twelfth
Standard: Data Analysis and Probability from the National Council of Teachers of Mathematics Principles and Standards for School Mathematics
Key Concept: Key Concept: Students are knowledgeable, analytical, thoughtful consumers of data.
Generalization: Students formulate a question that can be addressed with data and collect, organize, and display the data.
Background:
This lesson would be an end-of-course culminating activity and should be completed in groups consisting of two to four students.
Students choose a tier according to interest in a question and decide to use a survey, observational study, or experiment to answer the question. 
Directions for all the tiers are the same. Students determine a question they would like to answer, decide on a appropriate means for data collection, and prepare a presentation of the information to share with the class. The presentation should include a complete analysis of the data and the answer to the question of interest. However, a variety of presentation methods would be appropriate, e.g., a poster, a PowerPoint display, a written report, or a radio/TV show interview.
This lesson will take a number of days to complete as students will need time to decide on a question, collect the data, analyze the data, and prepare the presentation. You will also need 1-2 days for students to make their presentations.
This lesson is tiered in process and product according to interest.
The tiers could be based on the questions or the products. Those listed here represent the products produced. 
Tier 1: Poster Tier II: Power Point Tier III: Written report Tier IV: Radio/TV
Assessment: Assessment:
A rubric for each product should be based primarily on neatness, organization, accuracy of the information, and accuracy of the statistical analysis. The rubrics should be given to the students at the beginning the lesson since the decision on which product will be made after selecting a question.

35. Sample Model Lesson Framework
Whole
Math Message/Warm – Up
Lesson
Part
Student Activity
Group 1
Group 2
Group 3
Share Time

36. Misunderstandings vs. Reality
Differentiation is a set of instructional strategies.
It’s adequate for a district or school leader to tell or show teachers how to differentiate effectively.
Differentiation is something a teacher does or doesn’t do.
Differentiation is just about instruction.
Reality
Differentiation is a philosophy – a way of thinking about teaching & learning – it is a set of principles.
Learning to differentiate requires rethinking of one’s classroom practice through assessment and adjustment.
Most teachers in a classroom for one day DO pay attention to student variations, put very few proactively plan to address all student differences.
Differentiation is inseparable from a positive learning environment, curriculum, and flexible management.
Based on a recently published book co-authored by C. Tomlinson Leading and Managing a Differentiated Classroom 2010

37. http://www. louisianabelieves
Hyperlink to show the links to tasks at various websites preparation standards (prerequisite skills and possibly remedial work).

38. It’s well to remember that: THINGS TAKE TIME