When, Why, and How Should

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When, Why, and How Should We Differentiate Math
Instruction in the Middle School?
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Mathematics is no longer for the selected few.
All children must be expected to succeed in
mathematics. NCTM Curriculum Standards
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Whats Differentiated Instruction?
It is a chance to offer a variety of learning
options that address different levels, interest
and learning styles of diverse needs of students
in mixed ability classrooms.
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What has changed in mathematics that makes it
important to do this?
Because of the changing needs of society, and in
order to function in a technological world, all
our students must learn mathematics in the
broadest scope. NCTM Standard Addend Series.
Kindergarten Book, p IX.
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Because our student populations have changed, our
math expectation for all students have to change.
We need to teach in a way that truly makes
academic success available to every student
regardless of cultural differences, and we need
to adjust the way we present learning experiences
for our students.
http//www.smcm.edu/academics/EdStudy/d7-Proj/Proj
ects/ResearchSites/acbrowning/index.htm
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What do these changing expectations for
mathematics have to do with incorporating
differentiating instruction into classrooms ?
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How has this change in the goal for mathematics
impacted the role of teacher ?

Teachers who showed greatest ability to move
toward differentiated classrooms were inquirers
about students and saw schooling as an organic
enterprise in which disequilibrium or disturbance
was a catalyst for growth Tomlinson
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How will DI change classroom procedures?
When students are busy making up their own minds,
the role of the teacher shifts. This new focus
defines the teacher as one who is circulating,
redirecting, disciplining, questioning,
assessing, guiding, directing, fascinating,
validating, facilitating, moving, monitoring,
challenging, motivating, watching, moderating,
diagnosing, trouble-shooting, observing,
encouraging, suggesting, watching, modeling and
clarifying. http//members.shaw.ca/priscillathe
roux/teacherrole.html
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It is a different approach to teaching and one
that runs counter to the way many of us learned
to do and teach mathematics.
Realistically, student-centered learning can be
time-consuming and messy, efficiency will
sometimes argue for the teacher directed whole
group lessons. (Jamie McKenzie, The WIRED
Classroom)
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Designing a DI lesson is more involved than a one
size fits all lesson.
In a differentiated lesson The teacher should
think in terms of what students must learn rather
than what tasks they must complete.
Casteel Johnson
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In a thinking curriculum, students develop an
in-depth understanding of the essential concepts
and processes for dealing with those concepts,
similar to the approach taken by experts in
tackling their tasks. (Herman et al., 1992, p.17)
When questioning, problem-solving and
investigation become the priority classroom
activities, all students reap the benefits of
active mathematics participation.
http//members.shaw.ca/priscillatheroux/teacherrol
e.html
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When we differentiate we build upon
Prior knowledge Prior experiences
Culturally defined values and norms Biological
differences in cognitive development Home
environment Maturity level Self-efficacy
Culturally determined perceptions of school and
learning
http//www.smcm.edu/academics/EdStudy/d7-Proj/Proj
ects/ResearchSites/acbrowning/index.htm
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Problems of the Homogeneous Classroom 1) Unless
you are teaching only one child, there will
always be levels in experiences and readiness for
any topic. 2) It is impossible to address an
individuals needs. 3) There are very few
opportunities to change groups or levels.
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4) Teachers in these types of classrooms are at
risk of developing limiting expectations for
their students. (The Self-fulfilling Prophecy).
(Schiedewind Davidson, 2000) 5) There is not
proof that such classrooms lead to higher test
scores, or a greater acquisition of
knowledge. 6) Placement in a lower level track
can be quite damaging to a childs self-efficacy
http//www.smcm.edu/academics/EdStudy/d7Proj/Proje
cts/ResearchSites/acbrowning/Benefits.htm
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• In a Heterogeneous Classroom
• Students are exposed to a variety of types of
  people which adds to their social experiences
  (Watson, 1985).
• Labels and stigmas for students are more likely
  to be avoided (Watson, 1985).
• Teachers can develop ways of effectively meeting
  the needs of all students in a heterogeneous
  classroom.

http//www.smcm.edu/academics/EdStudy/d7-Proj/Proj
ects/ResearchSites/acbrowning/Benefits.htm
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• The differences between students in a
  heterogeneous classroom are loud and clear.
  Teachers will have to acknowledge and address
  these differences, unlike in homogenous
  classrooms where the assumption exists that one
  method of teaching will fit all students.

• In order to make learning in a heterogeneous
  classroom accessible to every student, a teacher
  must be willing to differentiate to a certain
  extent. Differentiation is challenging, but there
  are many benefits to both the teacher and the
  students.

http//www.smcm.edu/academics/EdStudy/d7-Proj/Proj
ects/ResearchSites/acbrowning/Benefits.htm
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Student Benefits of a Differentiated Classroom
Every student has an opportunity to succeed.
A single experience with success is enough for a
student to approach new learning situations with
confidence and motivation (Stronck, 1980).
Opportunity is there to discover personal
strengths and show multiple intelligences.
Less frustration due to confusion or boredom.
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Benefits to the Teacher More sense of control
over each students learning progress (Tomlinson,
1995). A greater understanding of each
students ability to learn. The reward of
having a classroom that allows equal opportunity
for success for all students.
http//www.smcm.edu/academics/EdStudy/d7-Proj/Proj
ects/ResearchSites/acbrowning/Benefits.htm
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Good planning takes effort and practice.
When you begin with a solid major concept
(sometimes called Big Idea or Enduring
Understanding) you will have an easier time
planning learning experiences that aim to enhance
understanding of that concept. Although the major
concept for every child is the same, you can
easily plan several ways to approach learning the
same concept.
http//www.smcm.edu/academics/EdStudy/d7-Proj/Proj
ects/ResearchSites/acbrowning/How/planning.htm
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The major concept (Big Idea or Enduring
Understanding) serves as an anchor for the
unit/lesson you are planning to differentiate.
Even if you decide to take different students on
varying paths of learning, they will all end up
at the same point with an understanding of the
same major concept.
http//www.smcm.edu/academics/EdStudy/d7Proj/Proje
cts/ResearchSites/acbrowning/How/planning.htm
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A key to differentiated instruction is assessment
prior to planning the details of a lesson. The
teacher who differentiates instruction plans his
or her lesson based on the information he or she
has gathered about what the students already
know, and still need to learn. The teacher can
then avoid planning a lesson that is way above or
below the students capabilities.
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Assessment
Many teachers may already differentiate
instruction to some extent by responding to the
needs of those learners who are either struggling
with a concept, or have already mastered the
concept being taught. However, this type of
microdifferentiation (Tomlinson 1995) occurs
after the teacher has already begun teaching a
single lesson to all of the students. This single
lesson was most likely planned without a variety
of opportunities for students to learn or express
knowledge, or with no plan for flexible grouping
based on preassessment.
http//www.smcm.edu/academics/EdStudy/d7-Proj/Proj
ects/ResearchSites/acbrowning/How/Assess.htm
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Content, Processes, and Products Content
refers to the concepts and sub-concepts each
student learns in a particular lesson.
Process refers to the learning experiences that
you choose to provide for your students to
achieve an understanding of the content in a
lesson. Products are the end result of the
lesson. Each student applies what she or he has
learned in the lesson to create a final product,
or to show his or her acquired skill.
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How can these elements of a lesson (content,
processes, and products) be different for
different students, and still allow each student
to come away with knowledge of the same major
concepts and skills in a 45 (ish) minute period?
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The Strategies for Differentiating
Instruction Adjusting Questions Readiness /
Ability Compacting Curriculum Learning
Contracts Tiered Lessons Learning Centers
Flexible Grouping Acceleration/Deceleration
Anchoring Activities Student Interest Peer
Teaching Learning Profiles/Styles
Buddy-Studies Independent Study
Projects
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Strategies to Differentiate
Adjusting Questions - Although everyone is
expected to acquire the same basic content the
level of complexity of the questions may differ.
Compacting Curriculum - This is based on
assessing who already knows what, and creating
relevant experiences for those who can move on.
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Tiered Lessons - adapting alternative ways to
reach a common goal. Flexible Grouping - allows
for creating appropriate challenges while
avoiding labels. This can encourage the
development of multiple leaders that arise in
different situations.
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Adjusting Question Complexity

• As you consider adjusting questions or group
  placement . .
• How does a student understand, define, or
• explain a task?
• How does a student organize their approach to the
• task?
• 3) Does the student see relationships?
• Does the student relate this work to similar
• problems?
• Does the student vary their approach to different
• problems?
• 6) Can they describe their strategy?
• 7) Does the student show evidence of thinking
  ahead or backwards?
• 8) Can the student generalize the process or
  results?
• 9) Can the student self-evaluate?
• 10) How does the student work in a
  group?

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• Curriculum Compacting is used when a teacher
  encounters a student who has already mastered a
  concept that other students have not.
• How is this done?
• Decide on a level of a achievement that
  indicates mastery. Many teachers use 80 on a
  preassessment as an indication of mastery (Reis,
  1998 Reis, 1992).
• Pre-assess students to decide who has already
  mastered the material.
• Plan enrichment activities or adjust content
  for learners who attain a mastery level.
• Use a record to keep track of the progress of
  the students.

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Common Difficulties with Curriculum Compacting
Trouble with designing enrichment
activities that are geared toward students
interests, and are quality. A lack of
support from experts on instruction for advanced
learners.
Using Curriculum Compacting to Challenge the
Above-Average, by Sally M. Reis and Joseph S.
Renzulli (1992).
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One way to differentiate is to focus on tiering
one lesson, rather that creating 3 separate
lesson for each concept. How can this be
accomplished fairly and reasonably?
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Flexible Grouping Be sure to keep groups from
becoming stagnant (Tomlinson, 2001). Students
need to work with a variety of peers in groups
based on interests, readiness, or self-selected
groups. Without changing the groups in your
classroom, students will quickly find a
relationship between how they are grouped and the
type of work they receive. Flexible groups are
also necessary because students abilities vary
from time to time. A student who is struggling
with one concept may excel in another (Tomlinson,
1995). Be sure to honor these changes.
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Flexible Groups Are decided upon in a variety
of ways, such as o Interests of
students o Readiness of students based on
pre- assessment data o Requests of
students
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Flexible Groups Allow students to decide
to leave a particular group if appropriate. (A
student may start off at a slow pace, but
progress at a quicker pace, so he or she may want
to move into the group with less guidance.)
Avoid labeling within a classroom.
Accommodate differences within an individual. A
single student may struggle with one concept, but
excel in another. Are always changing.
Students should not be able to predict what group
they will be assigned to.
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• Assessment is not Necessarily Evaluation
• Assessment should be defined simply as gathering
  information about students (Gregory Chapman,
  2002).
• Such information may include
• What do your students know about the concept you
  are
• teaching?
• What else do they need to learn?
• How well are they formulating an understanding of
  what
• you have taught so far?
• How did the student respond to this type of
  teaching style?

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The Locker Problem A school has 1,000 lockers
and 1,000 students. The students decide to have
fun one day, so they take turns opening and
closing the lockers, according to the following
plan. The first student opens every locker.
The second student closes every second locker.
The third student opens every third closed
locker. The fourth student closes every fourth
open locker. The students continue in this
manner until all 1,000 students have had their
turn. When all the students are finished, which
lockers remain open?
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A Quarter of Me Materials Large
paper String Markers (Crayons) Scissors Rulers
We are moving to a land of different people. To
live and to play the games there we need to be
proportionally smaller. For this land we need to
be a quarter of our size. Use the string to
measure your height, length of arms and legs,
width of your shoulders, the distance from the
top of your head to your shoulder, distance from
your shoulder to your waist, distance from your
waist to the floor, distance from ear to ear,
distance from hip to hip, and length of
foot. Using the string measurements, create a
proportional representation of your body on the
large paper.
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Work with a partner. Use the string to measure.
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As you start to plan . . . What is a good focus
question? What does the student need to know
about this unit? What should they be able to
do? Recognize Identify Define Review
Compare Classify Create Other ? What types of
questions should the student be able to
answer? What resources will you
use? Considering the time limits, what are
reasonable expectations for the unit? Connect to
the NCTM Standards - 2000
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With your team, explain how you would
differentiate this lesson in three different
levels.
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• Remember as you start to plan
• Theres no single perfect example of
• differentiated instruction.
• Differentiating takes effort, time, and is a
• career long pursuit.
• Start with a unit, concept, or lesson and try it
  out.
• Start by practicing forming good, broad essential
• questions and clear objectives of the lesson.

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• Take small steps Start with just one unit.
• Reflect on your progress.
• Communicate with your students and their
• parents about your decision.
• Seek out a colleague as support and for
• collaboration.

http//www.smcm.edu/academics/EdStudy/d7-Proj/Proj
ects/ResearchSites/acbrowning/Advice.htm
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As You Try to Differentiate Instruction, What
Goals Should You Aim for With Your Students?
Trust Make sure that you are open with your
student about your decisions. For example Why
have you chosen to put a students in a particular
group? High Expectations Always insist on
high quality work from every student. Explain why
something is or is not meeting your
expectations. A Community of Learners
Convey the message that every student is an
important part of your classroom. We are all in
this together.
http//www.smcm.edu/academics/EdStudy/d7-Proj/Proj
ects/ResearchSites/acbrowning/How/environ.htm
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• Differentiated Instruction Resources
• http//www.spots.ab.ca/ptheroux/differentiating.h
  tml
• Great description of differentiated instruction.
• http//www.kidsource.com/kidsource/content/diff_in
  struction.html
• A web resource provided by an expert on
  differentiated instruction--Carol Ann Tomlinson.
• http//www.teachnology.com/litined/dif_instruction
  /
• This site provides links to several articles on
  differentiated instruction

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Check out the following web sites for help with
developing strong Major Concepts/Enduring
Understandings http//xnet.rrc.mb.ca/glenh/und
erstanding_by_design.htm Backward Planning.
This method of planning was made popular by
http//www.ubdexchange.org/ Another resource
on planning from Understanding by Design.