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Differentiated Instruction In Math. Differentiated Instruction in Math. Schools That Make a Difference High quality teachers – with high expectations for all students Rigorous curriculum – involves conceptual engagement and productive struggle

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differentiated instruction in math2
Differentiated Instruction in Math

Schools That Make a Difference

  • High quality teachers – with high expectations for all students
  • Rigorous curriculum – involves conceptual engagement and productive struggle
  • Schools organized so students are known and well-supported
slide3

I like this class because there’s something different going on all the time. My other classes, it’s like peanut butter for lunch every single day. This class, it’s like my teacher really knows how to cook. It’s like she runs a really good restaurant with a big menu and all.

      • Comment from a course evaluation written by a 7th grader
what is differentiated instruction
What is Differentiated Instruction?
  • Differentiation means tailoring instruction to meet individual needs. Whether teachers differentiate content, process, products, or the learning environment, the use of ongoing assessment and flexible grouping makes this a successful approach to instruction.
      • Carole Ann Tomlinson
differentiated instruction in math5
Differentiated Instruction in Math

Achievement Gap or Instructional Gap?

How do differentiation and interventions make a difference?

  • Guided math
    • Fluid subsets of the entire class
    • Working with the core group and extra work in areas where students struggle to catch them up and keep them up
    • Enrichment and extensions
  • Extra time and extra instruction – THE CHALLENGE
  • Time and support must be the variables
    • Schools are developing strategies to provide students with that time to:
      • Experience the content with different approaches and representations
      • Have more time to discuss, reason, and process
differentiated instruction in math6
Differentiated Instruction In Math

Differentiated Instruction Addresses:

  • Relationships – knowing and valuing individual learners
  • Relevance – involves real-world problem solving, addresses student interests and learning styles
  • Rigor – challenges every student at their level of readiness
  • Response to Instruction/Intervention
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Differentiated Instruction In Math

2 Essentials

  • Knowing our learners
  • Knowing our curriculum
slide8
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Differentiated Instruction In Math

Celebrating Differences

  • No two children are alike. 
  • No two children learn in the identical way. 
  • An enriched environment for one student is not necessarily enriched for another. 
  • In the classroom we should teach children to think for themselves.

The Most Wonderful Egg in the World

      • http://fno.org/nov97/egg.html
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Differentiated Instruction In Math

Knowing Our Learners

  • Assessing Student Readiness
    • Formal assessments and inventories
    • Informal assessments and observations
    • Open-ended tasks
    • Ongoing formative assessments
    • Student conversations
    • Anecdotal student work
    • Questioning
    • KWL’s
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Differentiated Instruction In Math

Knowing Our Learners

  • Student Interests/Passions
    • Conferences
    • Journals
    • Interest Surveys
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Differentiated Instruction In Math

Knowing Our Learners

Student Learning Profiles

  • Are they . . . . . .

Left Brained? Right Brained? Intuitive? Sensing?

Feeling? Abstract? Concrete? Sequential?

Random? Visual? Auditory? Kinesthetic?

Global? Analytical?

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Differentiated Instruction In Math

Knowing Our Learners

What are their “Preferred Intelligences”?

  • Verbal/Linguistic Mathematical/Logical
  • Bodily/Kinesthetic Musical/Rhythmic
  • Intrapersonal Interpersonal
  • Visual/Spatial

Naturalist

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Differentiated Instruction In Math

Does it even matter?

  • Yes, chickens and eggs are important!!
  • Discover their learning styles (and your own!)
  • Multiple Intelligences http://surfaquarium.com/MI/inventory.htm
  • Learning Styles Global/Analytical Test and Modalities Inventory

http://www.berghuis.co.nz/abiator/lsi/lsiframe.html

  • Dr. Felder’s Inventories –Index of Learning Styles Test
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Differentiated Instruction In Math

Knowing Our Curriculum

  • Let’s examine the NGSSS for Mathematics
    • Florida Standards Database
  • Benchmark Comparisons
    • What is a mastery curriculum
    • Is it possible to have a mastery curriculum without differentiation?
elements of differentiated instruction
Elements of Differentiated Instruction

Teachers can differentiate at least four classroom elements based on student readiness, interest, or learning profile:

  • Content – what the student needs to learn or how the student will get access to the information;
  • Process – activities in which the student engages in order to make sense of or master the content;
  • Products – culminating projects that ask the student to rehearse, apply, and extend what he or she has learned in a unit; and
  • Learning environment – the way the classroom works and feels.
    • Carol Ann Tomlinson
principals of differentiated instruction
Principals of Differentiated Instruction
  • The teacher is clear about what is important in subject matter.
  • All students participate in respectful work.
  • The teacher understands, appreciates, and builds on students’ differences.
  • Assessment and instruction are inseparable.
  • The teacher adjusts content, process, and product in response to students’ readiness, interests, and learning profile.
  • Students and teachers are collaborators.
  • The goals of a differentiated classroom are maximum growth and individual success.
  • Flexibility is the hallmark of a differentiated classroom
differentiated instruction
Differentiated Instruction
  • The Basics

1. Responsive Learning Environment

      • How can I optimize the learning environment to maximize achievement for all students?
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Differentiated Instruction
  • The Basics

2. Clarity about the learning goals

      • What do my students need to know, understand, and be able to do?
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Differentiated Instruction
  • The Basics

3. Continuous Assessment

      • Pre-assessment
        • Quick checks
        • Benchmark assessments
          • www.florida-achieves.com
      • Ongoing Assessment (Formative)
        • Quick checks
        • Questioning strategies
        • Benchmark assessments
          • www.florida-achieves.com
        • Progress monitoring
      • Summative
        • Topic, Unit, Mid-year, End-of-Year, FCAT
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Differentiated Instruction
  • The Basics
    • Formative assessments before during and after instruction:
      • Benchmark mini-assessments
      • Response cards and boards
      • Exit slips
      • Surveys
      • Portfolios
      • Journal discussions
      • Student discussions – reasoning and justification
      • Performance assessment
      • Student-led conferencing
      • Student-to-teacher conference
      • Student-generated assessments
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Differentiated Instruction
  • The Basics
    • Flexible Grouping
      • Whole Group
      • Teacher-guided Groups
      • Individualized Instruction
      • Peer Tutoring
      • Class-wide Peer Tutoring
      • Cooperative Learning
      • Reciprocal Teaching
      • Independent Small Groups
      • Learning Centers
      • Cross-Age Tutoring
      • Mixed-Readiness Groups
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Differentiated Instruction
  • The Basics

4. Respectful Work

      • Addresses the knowledge, understandings, and skills the teacher has identified as important
      • Can be differentiated according to:
        • Readiness for what is being taught
        • Interests
        • Learning preferences
      • Can be created by varying:
        • Complexity/Difficulty
        • Amount of structure/Number of steps
        • Scaffolding provided
        • Timeline for completion
        • Pacing
        • Materials
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Students at work create various kids of noise. They talk and measure and puzzle out and make the audible messes that an assistant principal is supposed to abhor. Their activity also exposes the inconvenient truth that some kids do the work faster than others. The neat march over material that is possible when only the teacher sets the pace of the journey is no longer possible.

  • Theodore Sizer – Horace’s School
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Differentiated Instruction In Math

Guiding Questions

  • What is the math I want my student to learn?
  • What do my students already know? What is my evidence of this? How can I build on their thinking?
  • How can I expand access to this task or idea? Have I considered student readiness, interests, and learning styles?
  • How can I ensure that each student experiences challenges? (content and process)
  • How can I tier and/or scaffold learning to increase the likelihood of success? (process)
  • In what different ways can my students demonstrate their understanding? (product)
  • Are there student choices?
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Differentiated Instruction In Math

Differentiated Content

  • A change in the material being learned by a student
    • If the objective is for all students to subtract two digit numbers:
      • Some students work on guided practice with the teacher on subtraction situations (CGI) in computation
      • Some students subtract larger numbers in a problem-solving context
    • Games and Virtual Manipulatives offer a great way to differentiate content and process
      • enVision and Investigations Games
      • SuperSource Activities
      • http://nlvm.usu.edu/en/nav/vlibrary.html
      • http://mathwire.com
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Differentiated Instruction In Math

Differentiated Process

  • Refers to the way a student accesses material
    • Some students may explore fraction concepts using manipulatives or drawings at a learning center with a small group of students
    • Some students may explore fraction concepts independently using virtual manipulatives on the Internet
    • Some students may work with manipulatives in a small group directed by the teacher
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Differentiated Instruction In Math

Differentiated Product

The way in which a student demonstrates what he or she has learned

  • To demonstrate understanding of a geometric concept:
    • Some students may solve a problem set
    • Some students may build a model
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Differentiated Instruction In Math

Other Strategies

  • Stations. Using stations involves setting up different spots in the classroom where students work on various tasks simultaneously. These stations invite flexible grouping because not all students need to go to all stations all the time.
  • Agendas. These are personalized lists of tasks that a student must complete in a specified time, usually two to three weeks. Student agendas throughout a class will have similar and dissimilar elements.
  • Complex Instruction. This strategy uses challenging materials, open-ended tasks, and small instructional groups. Teachers move among the groups as they work, asking students questions and probing their thinking.
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Differentiated Instruction In Math

Let’s view a classroom using menus to facilitate differentiation and effective, problem based instruction at all levels of abstraction:

  • Concrete
  • Representational
  • Abstract

*Marilyn Burns DVD

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Differentiated Instruction In Math
  • Tiering Instruction
    • Allows students to focus on the same general concept or skill according to their level of readiness
    • Provides different levels of complexity, abstractness, and open-endedness
      • C-R-AWeb’s Levels/Depth of Knowledge
    • Activites can be tiered for difficulty or complexity or both
    • Provides appropriate challenges for all students
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Differentiated Instruction In Math

Tiering Instruction:

  • Can be based on content/input
  • Can be based on process/sense-making
  • Can be based on product/output

Based on student:

  • Readiness
  • Interests
  • Learning Style
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Differentiated Instruction In Math

A tiered lesson is a differentiation strategy that addresses a particular standard, key concept, and generalization, but allows several pathways for students to arrive at an understanding. Tomlinson (1999)

Creating Tiered Assignments

1. Identify the benchmark you are teaching

2. Identify the key concepts

3. Develop or select an activity that is interesting, requires high-level thought and causes students to work with the specified knowledge, understanding, and skill.

4. Determine the complexity level of the starting point task compared to the range of student readiness.

5. Develop multiple versions of the task at different levels of difficulty. Make sure all versions (levels 1,2,3) focus on essential knowledge, understanding, and skill.

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Differentiated Instruction In Math

Differentiation means doing something different—qualitatively different. Make sure you keep this in mind when tiering the lessons. Second, be sure that students are doing challenging, respectful, and developmentally appropriate work within each tier. In other words, no group should be given “busywork.” We don’t want one group doing backline practice sheets and another doing a fabulous investigation. Pierce & Adams

Creating Tiered Assignments – Continued

6. Pre-assess student understanding/mastery of key concepts/skills. Open-ended tasks work well.

7. Assign students to appropriate versions of the task.

8. Assess the students - The assessment can be formative, summative, or a combination of both – formal or informal. You may use some means of recording observations of the various groups, such as flip cards or sticky notes. You could develop a rubric for each tier based on the particular product that is created. You may give a formal paper-and-pencil test. Student conversations, writing, representations, and reasoning and justifications provide the most important and relevant insights into student learning.

  • The Tiered Curriculum Project – Math lessons tiered according to readiness, interests, and learning styles

http://www.doe.in.gov/exceptional/gt/tiered_curriculum/welcome.html

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Differentiated Instruction In Math

Measurement – Grade 2 – MA.2.G.3.1

Estimate and use standards units, including inches and centimeters, to partition and measure lengths of objects.

Key Ideas

Relationship between number of units and relative size of units

Pre-assess

Gather students with their journals around a table. Ask “How could we measure the length of this table with craft sticks?”

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Differentiated Instruction In Math

Measurement – Grade 2 – MA.2.G.3.1

Estimate and use standards units, including inches and

centimeters, to partition and measure lengths of objects.

  • Possible answers
    • Line them up - concrete
    • Use one and keep putting it down – less concrete – iterations of a unit
    • Measure the stick with a ruler. Use a ruler to measure length. – more abstract

Ask students to record their estimates of length in their journals.

  • Put 6 craft sticks end-to-end on the table’s edge, and give the students a chance to readjust their estimates.
  • Place craft sticks end-to-end to measure the entire table.
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Differentiated Instruction In Math

Measurement – Grade 2 – MA.2.G.3.1

Estimate and use standards units, including inches and

centimeters, to partition and measure lengths of objects.

Ask the students to draw what they think is a one inch line in their journals.

  • Show the students the length of an inch and ask them to estimate in their journals how long the table is in inches.
  • Explore students problem solving attempts.

Differentiate tasks - Content

  • Some students measure more lengths in order to better conceptualize the length of an inch.
  • Some students measure straight paths/lengths.
  • Some students measure crooked paths.
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Differentiated Instruction In Math

Measurement – Grade 2 – MA.2.G.3.1

Estimate and use standards units, including inches and

centimeters, to partition and measure lengths of objects

Differentiate Tasks – Process

  • Some students make inch and foot strips and use these models to help in estimation.
  • Length measures are posted around the room to provide visual images of length.
  • Inch and foot lengths of Velcro are placed on the wall for students to run their fingers over.
  • Some students are provided multiple one inch strips; others have only one which they must use over and over.
  • Some students measure using standard rulers or yardsticks/meter sticks.
  • Initially, students choose lengths they want to estimate.
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Differentiated Instruction In Math

Measurement – Grade 2 – MA.2.G.3.1

Estimate and use standards units, including inches and

centimeters, to partition and measure lengths of objects.

Differentiate Tasks – Product

  • Estimate ribbon lengths for an art project
  • Participate in an Estimation Olympics
  • Teach a younger student about estimation
  • Write an explanation of the estimation process
  • Pretend they’re interviewing for an estimation job and explain how their skills and experiences will help them in their job
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Differentiated Instruction In Math

Attributes of a Worthwhile Task

It is developmentally appropriate. – Ideas are within reach from a cognitive and experiential level. Concrete manipulatives and drawing materials should always be available to support student work.

(readiness, rigor)

It is contextualized. – Present ideas in connection to their real world, science, social studies, literature, and art. (interests, relevance)

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Differentiated Instruction In Math

Attributes of a worthwhile task

It offers an appropriate level of challenge. – A task must offer a cognitive challenge that requires decisions to be made and new ideas to be explored. When a task has multiple entry points, it is possible to engage students with a broad range of readiness. (readiness, rigor)

It encourages multiple perspectives. – An interesting task stimulates a variety of strategies, representations, and mathematical ideas and encourages mathematical discourse in which students explain and justify their thinking. (rigor, relevance)

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Differentiated Instruction In Math

Tiering – Third Grade Fractions

Multiple versions of a task providing equitable access to understanding.

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Differentiated Instruction In Math

Kindergarten MA.K.A.1.1

Represent quantities with numbers up to 20,

verbally, in writing, and with manipulatives.

Ordering Numbers: Red

Materials: 3 paper plates, counters, and number cards 0-6

Task:

Create a specific set of objects based on quantity.

Turn over a number card and make a set of objects to show how many.

Keep going until you have made 3 different sets.

Which set has the most?

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Differentiated Instruction In Math

Kindergarten MA.K.A.1.1

Represent quantities with numbers up to 20,

verbally, in writing, and with manipulatives.

Ordering Numbers: Blue

Materials: 6 paper plates, counters, and number cards 0-6

Task:

Create a specific set of objects based on quantity.

Turn over a number card and make a set of objects to show howmany.

Keep going until you have made 6 different sets.

Put the sets in order.

Which set has the most?

Which set has the least?

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Differentiated Instruction In Math

Kindergarten MA.K.A.1.1

Represent quantities with numbers up to 20,

verbally, in writing, and with manipulatives.

Ordering Numbers: Green

Materials: 6 paper plates, counters, and number cards 0-10

Task:

Create a specific set of objects based on quantity.

Turn over a number card and make a set of objects to show how many.

Keep going until you have made 6 different sets.

Put the sets in order.

Which set has the most?

Which set has the least?

Which sets are missing?

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Differentiated Instruction In Math

First Grade MA.1.A.4.1

Extend repeating and growing patterns, fill in missing terms,and justify reasoning.

Trains

Second Grade MA.2.A.2.2

Add and subtract multi-digit whole numbers through three digits with fluency by using a variety of strategies, including invented and standard algorithms and explanations of those procedures.

Sums Investigations Red

Sums Investigations Blue

Sums Investigations Green

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Differentiated Instruction In Math

Third Grade MA.3.A.1.1

Model multiplication and division including problems

presented in context: repeated addition, multiplicative

comparison, array, how many combinations,

measurement,and partitioning.

The Game of Leftovers

Fourth Grade MA.4.A.4.1

Generate algebraic rules and use all four operations to

describe patterns, including nonnumeric growing or

repeating patterns.

Arch Patterns Red

Arch Patterns Blue

Arch Patterns Green

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Differentiated Instruction In Math

Fifth Grade MA.5.S.7.1 & MA.5.S.7.2

Construct and analyze line graphs and double bar graphs. Differentiate between continuous and discrete data and determine ways to represent those using graphs and diagrams.

Data Collection and Analysis Investigation

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Differentiated Instruction In Math

The Role of Literacy

The relationship between language and mathematics is significant.

  • In teaching problem centered math, math tasks are often presented within language-rich contexts.
  • Story Problems

Old – Eight birds . .

New – After the rain there were eight birds playing in the puddle.

  • “Show your work” has been replaced with “explain your thinking.” (math process skills – communication, reasoning and justification)
  • Literature captures the heart of children and provides a context for learning math.
  • Literature provides mathematical connections to the real world.
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Differentiated Instruction In Math

Tiered Instruction K-2 – Growing and Repeating Patterns

MA.K.A.4.1 Identify and duplicate simple number and non-numeric repeating and growing patterns.

MA.1.A.4.1 Extend repeating and growing patterns, fill in missing terms, and justify reasoning.

MA.2.A.4.3 Generalize numeric and non-numeric patterns using words and tables.

Illuminations – Creating, Describing, and Analyzing Problems

http://illuminations.nctm.org/LessonDetail.aspx?id=U71

In this cluster of activities, students use interactive math applets to learn about repeating and growing patterns.

  • In the first part, students use an interactive square-pattern grid to create, describe, and analyze repeating patterns based on a two-square pattern unit.
  • In the second part, repeating patterns with pattern units of three, four, and five squares are investigated.
  • In Part 3, students use an interactive cube-sequencing applet to analyze repeating patterns of colored cubes.
  • In Part 4, growing patterns of colored cubes are created, described, and analyzed, and compared to repeating patterns.
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Differentiated Instruction In Math
  • Scaffolding Instruction
    • Provides temporary supports to allow students to accomplish tasks
      • Manipulatives, hundreds charts, graphic organizers, word walls, representations
    • Assists the student toward independence
    • Relates to Vygotsky’s zone of proximal development
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Differentiated Instruction In Math

Strategies for Scaffolding

Questioning – Effective questioning strategies require students to move beyond their comfort level to more complex thinking

  • Focus on questions that require more than simple recall
  • Ask questions at all levels of Bloom’s Taxonomy
    • Remember that Bloom’s taxonomy is fluid – your questioning does not always have to be linear
    • Scaffolding questions helps students reach more sophisticated levels of cognition
    • *Scaffolds are appropriate for our above as well as on and below level students.
  • Help students develop mathematical thinking with effective questions (see handouts)
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Differentiated Instruction In Math

Scaffolding Instruction

Scaffolds for Vocabulary

  • Word Walls
  • Word Banks - see the Florida Standards Database to download vocabulary
  • Graphic Organizers with varying levels of support
  • Concept Maps
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Differentiated Instruction In Math

Scaffolding Instruction

  • Graphic Organizers
    • Venn Diagrams
    • Graph Paper
    • Grids
    • Dot Paper
    • Triangle Pattern Block Paper
    • Inspiration/Kidspiration – flow charts, concept maps
differentiated instruction in math57
Differentiated Instruction In Math

Scaffolding Instruction

Rolling for Nets

Students create 2 dimensional nets from 3 dimensional shapes by rolling the 3-D shapes on a paper and tracing each face of the shape when it rests on the paper.

How might this task be scaffolded?

differentiated instruction in math58
Differentiated Instruction In Math

Scaffolding Instruction

Rolling for Nets

How might this task be scaffolded?

  • Possible scaffolds:
    • Teacher modeling of process
    • Larger shapes for some students
    • Tiered progression of shapes for student groups from cubes through cylinders and cones
    • Grid paper to record nets
    • Stickers to place on faces copied
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Differentiated Instruction In Math

Scaffolding Instruction

  • It is important to remove scaffolds as students progress.
  • The goal of scaffolding is for the problem solving process to be organized by the student.
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Differentiated Instruction In Math

Compacting Instruction - for more ready learners

    • Strategy recognizes that content can be accelerated or eliminated for some learners

Compacting requires

  • Identifying key curricular ideas
  • Pre-assessing students and making learning decisions based on the data
  • Designing activities to meet student readiness
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Differentiated Instruction In Math

Compacting Instruction – Differentiating for Advanced/Gifted Students

Mathematics Investigation Centers – focus on mathematics investigations related to reading, writing, science, social studies, logic and reasoning, data collection and analysis, and problem solving.

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Differentiated Instruction In Math

Choices – Students should become experts on their own learning needs

Embedded Choices

1. Content

  • Vary the numbers in story problems
    • The Monkey Bar Problem – Students are challenged to cross the decade number 10.

Students are presented a story problem with

two sets of number choices.

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Differentiated Instruction In Math

Choices – Students should become experts on their own learning needs

Embedded Choices

2. Solution Process/Strategy Choice

  • Allow students to experiment with their own strategies for problem solving
    • Direct Modeling
    • Counting
    • Derived Facts
  • Encourage students to choose to take risks and experiment with more sophisticated strategies when basic strategies are no longer challenging or efficient.
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Differentiated Instruction In Math

Choices – Students should become experts on their own learning needs

Embedded Choices

3. Working Conditions

  • Students self –select from varying levels of support based on their level of comfort with a task
    • Work independently
    • Work with a partner
    • Work with an adult
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Differentiated Instruction In Math

Other Choices :

  • Topics to study
  • Assessments (student-generated)
  • Tasks to complete
  • Partners
  • Materials to use
  • Where to work
  • Order in which to complete tasks
  • Media to present ideas and research
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Differentiated Instruction In Math

Choices

  • Math Menus
    • Main Courses – focus on key instructional objectives (standards)- must be completed
    • Side Dishes – reinforce key instructional objectives
    • Desserts – challenges, extensions, or enjoyable activities
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Differentiated Instruction In Math

Choices

  • Learning Stations/Differentiated Activities
    • To supplement or support present skill or concept development
    • As a review
    • As extensions or enrichments
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Differentiated Instruction In Math
  • How do you currently differentiate to meet differing student needs, interests, and learning styles?
  • Do you plan independently or with a grade level team?
  • What do you see as the greatest challenge to differentiation in math (other than time)?
  • What do you feel will be the greatest advantages to differentiation?
differentiated instruction in math69
Differentiated Instruction In Math
  • Essential Ingredients
    • TEACHER COLLABORATION!
    • Energetic, enthusiastic, committed teachers who believe in the “genius” of every child
    • Classrooms organized into small groups – redefines student responsibilities and the teacher’s role
    • Flexible groups based on ongoing embedded assessment
    • Activities which address student readiness, learning profiles, and interests
    • Choices for students
    • Relevant and meaningful problem solving opportunities, opportunities to contribute
    • Multiple resources
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Differentiated Instruction In Math
  • Getting Started: Managing the Classroom
    • Rules of Engagement for Students Marilyn Burns
      • You are responsible for your own work and behavior. The six inch rule - voices
      • You must be willing to help any group member who asks.
      • You may ask the teacher for help only when everyone in your group has the same question. Ask 3 before me:-}
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Differentiated Instruction In Math
  • Instructional Resources
    • enVision and Investigations http://www.pearsonsuccessnet.com
    • About Teaching Mathematics – Marilyn Burns
    • Writing in Math Class - Marilyn Burns
    • Math Literature (Marilyn Burns Libraries) – see CD for further literature resources
    • Teaching Student Center Mathematics K-3 and 3-5 and 5-8 by Van de Walle
    • The Super Source CD - ETA Cuisenaire
    • Hands-On Standards PreK-K, 1-2, 3-4 & 5-6
    • Technology
    • Math Games
    • St. Johns County Curriculum Maps https://inside.stjohns.k12.fl.us/Login/__www/depts/cl/is/
slide72

I find the great thing in this world is not

so much where we stand, as in what

direction we are moving.

Oliver Wendell Holmes

Taking the first steps . . .