Universal Design for Learning
CCSS for Mathematics
Kitty Rutherford and Mary Keel
http://wikicentral.ncdpi.wikispaces.net
Click on Region 2 (bottom right)
CCSA UDL Math
AGENDA
• Recognizing
g
g Connections between learning
g and
neuroscience
• Understanding
g the three UDL p
principles
p
• Reviewing examples of math practice that illustrate
alignment of UDL to curriculum
• Discovering hands-on exploration in math that support
UDL
• Clarifying the curriculum framework as a structure for
designing lessons
• Resources for Next Steps
“N
“Norms”
”
• Listen as an Ally
• Value Differences
http://thebenevolentcouchpotato.wordpress.com/201
1/11/30/norm-peterson-bought-the-house-next-door/
• Maintain Professionalism
• Participate Actively
Parking Lot
http://wallwisher.com/wall/gt6xelyr8x
5
“Teachers must …regard
every imperfection in the
pupil’s
il’ comprehension
h
i nott
as a defect in the pupil, but
as a deficit in their own
instruction, and endeavor
to develop
p the ability
y to
discover a new method of
teaching.”
–Leo Tolstoy
Instead of saying
“students can’t”,
we now identify
instructional strategies
that demonstrate
“how students can”.
What is Universal Design for
Learning?
Universal Design for Learning
A universally
i
ll d
designed
i
d
curriculum is
d
developed
l
d ffrom th
the
start to be accessible
as wellll as challenging,
h ll
i
for ALL students.
UDL has its basis in
neuroscience
Three principles correlate with the three
networks in the brain:
• Recognition Network
• Strategic
St t i Network
N t
k
• Affective Network
The three must be simultaneously engaged for optimal learning to occur.
Recognition Networks
• Gathering facts. How we identify and
categorize what we see
see, hear
hear, and read
read.
• Identifying
y g letters, words, or an author's
style are recognition tasks
the ""what"
hat" of learning.
learning
Strategic Networks
• Planning and performing tasks.
• H
How we organize
i and
d express our id
ideas.
Writing an essay or solving a math
problem
bl
are strategic
t t i ttasks—
k
the "how"
how of learning
Affective Networks
• How students are engaged and
motivated.
• How they are challenged, excited, or
i t
interested.
t d These
Th
are affective
ff ti
dimensions
the "why" of learning
We have talked about the
three primary brain networks…
What should be some
considerations when
developing plans for your
classroom?
Three UDL Principles
A universally-designed
universally designed curriculum offers:
• Multiple means of representation to give learners
various ways of acquiring information and knowledge
• Multiple means of action and expression to provide
learners alternatives for demonstrating what they know
• Multiple means of engagement to tap into learners'
interests challenge them appropriately
interests,
appropriately, and motivate
them to learn
Multiple Means of
Representation
• The “what” of learning
• Present information and content in
different ways
Multiple Means of Action
and Expression
• The “how” of learning
• Differentiate the ways the students can
p
what they
y know
express
Multiple Means of
Engagement
• The “why” of learning
• S
Stimulate interest and motivation ffor
learning
What is Universal Design for Learning?
- a set of principles for curriculum
development that applies to the general
education curriculum that gives all
individuals equal opportunities to learn.
Universal Design for Learning
provides a blueprint for creating
instructional goals, methods, materials, and
assessments that work for everyone--not a
single one-size-fits-all solution but rather
single,
flexible approaches that can be customized
and adjusted for individual needs.
Universal Design for Learning
Purpose of UDL Curriculum
is not simply to help students master a
specific
p
body
y of knowledge
g or a specific
p
set of skills, but to help them master
learning
g itself—in short,, to become expert
p
learners.
Let’s
L
t’ think
thi k about
b t some math
th
considerations when
developing UDL plans for
division
• Discuss at your table
• Share
Sh
your id
ideas on W
Wallwisher
ll i h
http://wallwisher.com/wall/hxqpnwkxac
•
Write down three
g that yyou
things
think are critical for
t
teaching
hi di
division.
i i
Research
Simply being able to perform calculations does
not necessarily mean that students understand
these operations. Conceptual knowledge is
based on understanding
g relationship
p between
multiplication and division. Since everyday
mathematics is almost always applied in the
context
t t off words,
d nott symbols,
b l it is
i important
i
t t
for students to understand the relationship
inherent in multiplication and division
problems.
How would you define division?
Common Core State Standards
Third Grade
Operations & Algebraic Thinking
Represent and solve problems involving multiplication
and division.
3.OA.2 Interpret whole-number
whole number quotients of whole numbers,
e.g., interpret 56 ÷ 8 as the number of objects in each
share when 56 objects are partitioned equally into 8
shares or as a number of shares when 56 objects are
shares,
partitioned into equal shares of 8 objects each. For
example, describe a context in which a number of shares
or a number of groups can be expressed as 56 ÷ 8.
Two Types
yp of Division
Partitive and Quotitive
Partitive (number in a group) division problems is one
of dividing or partitioning a set into a predetermined
number of groups.
Twenty-four apples need to be placed into eight paper
sacks. How many apples will you put in each sack if
you want the same number in each sack?
If students use partitive division problems exclusively in
instruction students often have difficulty making sense
instruction,
of quotitive/measurement division problems.
In quotitive/measurement (number of groups) division
problems
bl
((also
l sometimes
ti
referred
f
d tto as repeated
t d subtraction
bt ti
problems) the number of objects in each group in known, but
the number of groups is unknown
For example:
F
l I have
h
24 apples.
l
How
H
many paper sacks
k will
ill I
be able to fill if I put 3 apples into each sack?
The action involved in quotitive/measurement (number of
groups)) division is one subtracting out predetermined
amounts. If asked to model this problem, students usually
repeatedly subtract 3 objects from a group of 24 objects and
then count the number of groups the removed (24 objects into
3 groups).
Students benefit from exposure to both types of division
examples so that they internalize that two actions
actions, subtracting
and partition, are used to find quotients.
Which type of multiplication is
most prevalent in the
classroom?
• Partitive (number in a group)
or
• Quotitive (number of groups)
Which type of Division
Partitive or Quotitive?
Max the monkey loves bananas. Molly
his trainer, has 24 bananas. If she
gives Max 4 bananas each day, how
many days will the bananas last?
video clip
Max the monkey loves bananas
bananas. Molly
his trainer, has 24 bananas. If she
gives Max 4 bananas each day,
day how
many days will the bananas last?
• How would you describe students’
strategies?
• What does your description indicate
about his or her understanding of division
and/or multiplication
Common Core State Standards
Third Grade
Operations & Algebraic Thinking
Represent and solve problems involving multiplication
and division.
3.OA.2 Interpret whole-number
whole number quotients of whole numbers,
e.g., interpret 56 ÷ 8 as the number of objects in each share
when 56 objects are partitioned equally into 8 shares, or as
a number of shares when 56 objects are partitioned into
equal shares of 8 objects each. For example, describe a
context in which a number of shares or a number of groups
can be expressed as 56 ÷ 8.
Max the monkey loves bananas
bananas. Molly
his trainer, has 24 bananas. If she
gives Max 4 bananas each day,
day how
many days will the bananas last?
Arrays in third grade helps students to
make the connect with multiplication and
division
Arrays in third grade making that connect to
multiplication and division
Repeated division with place value blocks
Max the money
y loves bananas. Molly,
y, his
trainer, has 24 bananas. If she gives Max
y how many
y days
y will the
4 each day,
bananas last?
Max the monkey loves bananas
bananas. Molly
his trainer, has 24 bananas. If she
gives Max 4 bananas each day,
day how
many days will the bananas last?
The action involved in quotitive/
measurement (number of groups)
division is one subtracting out
predetermined amounts. Student need
this experience to build understanding
Max
a tthe
e monkey
o ey loves
o es ba
bananas.
a as Molly,
o y,
his trainer, has 24 bananas. If she
gives Max 4 bananas each day,
g
y, how
many days will the bananas last?
• H
How would
ld you d
describe
ib students’
t d t ’
strategies?
• What does your description indicate
about his or her understanding of division
and/or multiplication
How have you seen the
principals of UDL
demonstrated?
• Discuss at your table
• Share your ideas on Wallwisher
http://wallwisher.com/wall/hxqpnwkxac
Which type of Division
Partitive or Quotitive?
Mrs. Campbell is arranging transportation for a
class trip
trip. She plans to drive
drive, and some parents
will too. Mrs. Campbell has 24 students in her
class, and she plans to assign 4 children to each
car How many cars will Mrs
car.
Mrs. Campbell need for
the trip?
Video Clip
Mrs. Campbell is arranging transportation for
a class trip. She plans to drive, and some
parents will too. Mrs. Campbell has 24
students
t d t in
i her
h class,
l
and
d she
h plans
l
tto
assign 4 children to each car. How many
cars will Mrs
Mrs. Campbell need for the trip?
• How would you describe students’
strategies?
t t i ?
• What does yyour description
p
indicate about
his or her understanding of division and/or
multiplication
Turn and Talk
Work with your table partners to decide if
the tasks are:
Group Size Unknown (Partitive)
or
Number of Groups Unknown
(Quotitive/Measurement)
Group Size or
Number of Groups Unknown
• A loaf of bread has 18 slices.
slices Mike’s
Mike s mom uses 6 slices
each time she packs lunches for the family. How many
times will she be able to make lunches from one loaf of
b d?
bread?
• Kevin has $15.00 to use to buy balls that cost $3.00
apiece How many balls can Kevin buy?
apiece.
• Katy is decorating goody bags for her birthday party.
She has 5 goody bags that she must decorate in the
next 35 minutes. How many minutes should she spend
on each bag?
Which examples
p
do most
teachers provide for students in
their classroom?
On chart paper write a few problems
using the quotitive/measurement
(number of groups) division problems
(also sometimes referred to as repeated
subtraction problems) the number of
objects in each group in known, but the
number of groups is unknown.
Common Core State Standards
Third Grade
Operations & Algebraic Thinking
Represent and solve problems involving multiplication and division.
3 OA 2 Interpret whole-number quotients of whole numbers,
3.OA.2
numbers e
e.g.,
g interpret 56 ÷ 8 as the number of
objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares
when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context
in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
Fourth Grade
Number & Operations in Base Ten¹
Use place value understanding and properties of operations to
perform multi-digit arithmetic.
4.NBT.6 Find whole-number quotients and remainders with up to fourdigit dividends and one-digit divisors, using strategies based on place
value, the properties of operations, and/or the relationship between
multiplication
lti li ti and
d di
division.
i i
Ill
Illustrate
t t and
d explain
l i th
the calculation
l l ti b
by
using equations, rectangular arrays, and/or area models.
Number in a Group
Using 4-digit by 1-digit
Mrs. Campbell’s class collected 3,468 cans
of food for the 3 food shelters
shelters. If her
students divide the cans evenly among the
shelters how many cans of food would each
shelter get?
How might a number of groups problem look?
Algorithms for Division
The long division algorithm is often difficult
f students
for
t d t to
t use and
d understand.
d t d
However, when teachers present an
abbreviated
bb i t d fform students’
t d t ’ understanding
d t di
is often sacrificed. Students demonstrate
l
less
proficiency
fi i
iin carry outt th
the algorithm
l ith
and make more errors.
NCCTN Developing Essential Understanding of
Multiplication and Division
Compounding the difficultly of division
notation is the unfortunate phrase, “six goes
into twenty-four.” This phrase carries little
meaning about division
division, especially in
connection with fair-sharing or partitioning
context. The “goes into” (or guzinta”)
terminology
i l
iis simply
i l engrained
i d iin adult
d l
parlance and has not been in textbooks for
years. If you tend to use that phrase, it is
probably a good time to consciously
abandon it.
Teaching Student-Centered Mathematics Grades 3-5
John Van de Walle
Now you try a problem using
an area model.
Mrs. Campbell’s
Campbell s class collected 3,468
cans of food for the 3 food shelters. If her
students divide the cans evenly among
the shelters how many cans of food
ou d eac
each sshelter
e te get
get?
would
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
others.
others
4. Model with mathematics.
5 Use appropriate tools strategically
5.
6. Attend to precision.
7 Look
7.
L k ffor and
d make
k use off structure.
t t
8. Look for and express regularity in repeated reasoning.
How have you seen the
principals of UDL
demonstrated?
• Discuss at your table
• Share your ideas on Wallwisher
http://wallwisher.com/wall/hxqpnwkxac
Common Core State Standards
Third Grade
Operations & Algebraic Thinking
Represent and solve problems involving multiplication and division.
3 OA 2 Interpret whole-number quotients of whole numbers,
3.OA.2
numbers e
e.g.,
g interpret 56 ÷ 8 as the number of
objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares
when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context
in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
Fourth Grade
Number & Operations in Base Ten¹
Use place value understanding and properties of operations to perform multi-digit arithmetic.
4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit
divisors, using strategies based on place value, the properties of operations, and/or the relationship
between multiplication and division. Illustrate and explain the calculation by using equations,
rectangular arrays, and/or area models.
Fifth Grade
Number & Operations in Base Ten¹
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5 NBT 6 Find whole-number
5.NBT.6
hole n mber quotients
q otients of whole
hole n
numbers
mbers with
ith up
p to fo
four-digit
r digit dividends
di idends and ttwo-digit
o digit
divisors, using strategies based on place value, the properties of operations, and/or the relationship
between multiplication and division. Illustrate and explain the calculation by using equations,
rectangular arrays, and/or area models.
An Area Model for Division
• Picture of division place value blocks
Division with Decimals
0 80 ÷ 0.20
0.80
0 20
0 30 ÷ 0.05
0.30
0 05
Research
The national Council of Teachers of
M th
Mathematics
ti recommends
d th
thatt students
t d t should
h ld
“develop a stronger understanding of various
meanings of multiplication and division
division,
encounter a wide range of representations and
problems situations that embody
p
y them,, learn
about the properties of these operations, and
gradually develop fluency in solving
multiplication and division problems.”
((NCTM 2000,, 149))
Educational Approach with 3 Primary
Pi i l
Principles
R i
Review
off Your
Y
Ideas
Id
• How did you see the Three Principles of
UDL demonstrated in the math lesson?
• Discuss at your table
• Share your ideas from Wallwisher
http://wallwisher.com/wall/hxqpnwkxac
Discussion
• What are the benefits of analyzing the
curriculum for strengths
g
and weaknesses
rather than focusing on the student’s
strengths
g
and weaknesses? What are the
challenges of this approach?
“Teachers must …regard
every imperfection in the
pupil’s comprehension not
as a defect in the pupil, but
as a deficit in their own
instruction, and endeavor
to develop the ability to
discover a new method of
teaching.”
–Leo Tolstoy
Instead of saying
“students can’t”,
we now identify
instructional strategies
that demonstrate
“how students can”.
Next Steps
• What are your next steps to integrate
UDL into yyour school environment?
http://cast.org/
R f
References
•
CAST, Inc: http://udlonline.cast.org
•
Rose, D., & Meyer, A. (2002). Teaching every student in the digital age:
Universal design for learning. Retrieved from
http://www.cast.org/teachingeverystudent/ideas/tes
•
/
http://aim.cast.org/learn/historyarchive/backgroundpapers/differentiated_in
struction_udl
DPI Contact Information
Kitty Rutherford
Elementary Mathematics Consultant
919 807 3934
919-807-3934
kitty.rutherford@dpi.nc.gov
Mary Keel
Professional Development Consultant
252 725 2570
252-725-2570
mary.keel@dpi.nc.gov
http://www wikicentral ncdpi wikispaces net
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