UDL and CCSS in Math

Universal Design for Learning
CCSS for MathematicsCCSS for Mathematics
Kitty Rutherford and Mary KeelKitty Rutherford and Mary Keel

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Click on Region 2 (bottom right)
CCSA UDL MathCCSA UDL Math

AGENDAAGENDA
• Recognizing Connections between learning andg g g
neuroscience
• Understanding the three UDL principlesg p 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
il’ h i tpupil’s comprehension not
as a defect in the pupil, but
as a deficit in their ownas a deficit in their own
instruction, and endeavor
to develop the ability top y
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 LearningUniversal Design for Learning
A i ll d i dA universally designed
curriculum is
d l d f thdeveloped from the
start to be accessible
ll h ll ias well as challenging,
for ALL students.

UDL has its basis inUDL has its basis in
neuroscience
Three principles correlate with the three
networks in the brain:
• Recognition Network
St t i N t k• Strategic Network
• Affective NetworkAffective Network
The three must be simultaneously engaged for optimal learning to occur.

Recognition Networks
• Gathering facts. How we identify and
categorize what we see hear and readcategorize what we see, hear, and read.
• Identifying letters, words, or an author'sy g
style are recognition tasks
the " hat" of learningthe "what" of learning.

Strategic Networks
• Planning and performing tasks.
H i d id• How we organize and express our ideas.
Writing an essay or solving a math
bl t t i t kproblem are strategic tasks—
the "how" of learningthe how of learning

Affective Networks
• How students are engaged and
motivated.
• How they are challenged, excited, or
i t t d Th ff tiinterested. These are affective
dimensions
the "why" of learning

We have talked about the
three primary brain networks…
What should be some
considerations when
developing plans for yourdeveloping plans for your
classroom?

Three UDL PrinciplesThree UDL Principles
A universally designed curriculum offers:A universally-designed curriculum offers:
• Multiple means of representation to give learners
various ways of acquiring information and knowledgevarious ways of acquiring information and knowledge
• Multiple means of action and expression to provide
learners alternatives for demonstrating what they knowlearners alternatives for demonstrating what they know
• Multiple means of engagement to tap into learners'
interests challenge them appropriately and motivateinterests, challenge them appropriately, and motivate
them to learn

Multiple Means ofMultiple Means of
Representation
• The “what” of learning
• Present information and content in
different ways

Multiple Means of ActionMultiple Means of Action
and Expression
• The “how” of learning
• Differentiate the ways the students can
express what they knowp y

Multiple Means ofMultiple Means of
Engagement
• The “why” of learning
S f• Stimulate interest and motivation for
learning

What is Universal Design for Learning?
- a set of principles for curriculum
development that applies to the general
education curriculum that gives alleducation curriculum that gives all
individuals equal opportunities to learn.

provides a blueprint for creating
instructional goals, methods, materials, and
assessments that work for everyone--not a
single one-size-fits-all solution but rathersingle, one-size-fits-all solution but rather
flexible approaches that can be customized
and adjusted for individual needs.

Purpose of UDL Curriculum
is not simply to help students master a
specific body of knowledge or a specificp y g p
set of skills, but to help them master
learning itself—in short, to become expertg , p
learners.

L t’ thi k b t thLet’s think about some math
considerations when
developing UDL plans for
divisiondivision
• Discuss at your table
Sh id W ll i h• Share your ideas on Wallwisher
http://wallwisher.com/wall/hxqpnwkxac
•

Write down threeWrite down three
things that youg y
think are critical for
t hi di i iteaching division.

Research
Simply being able to perform calculations does
not necessarily mean that students understandnot necessarily mean that students understand
these operations. Conceptual knowledge is
based on understanding relationship betweeng p
multiplication and division. Since everyday
mathematics is almost always applied in the
t t f d t b l it i i t tcontext of words, not symbols, it is important
for students to understand the relationship
inherent in multiplication and divisioninherent 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 quotients of whole numbers,3.OA.2 Interpret 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 areshares, 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.

Two Types of Divisionyp
Partitive and Quotitive
Partitive (number in a group) division problems is one
of dividing or partitioning a set into a predetermined
number of groups.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 senseinstruction, students often have difficulty making sense
of quotitive/measurement division problems.

In quotitive/measurement (number of groups) division
bl ( l ti f d t t d bt tiproblems (also sometimes referred to as repeated subtraction
problems) the number of objects in each group in known, but
the number of groups is unknown
F l I h 24 l H k ill IFor example: I have 24 apples. How many paper sacks will 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 intothen 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 subtractingexamples so that they internalize that two actions, subtracting
and partition, are used to find quotients.

Which type of multiplication is
most prevalent in themost prevalent in the
classroom?
• Partitive (number in a group)
or
• Quotitive (number of groups)• Quotitive (number of groups)

Which type of DivisionWhich 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?

Max the monkey loves bananas MollyMax the monkey loves bananas. Molly
gives Max 4 bananas each day howgives Max 4 bananas each day, how
• How would you describe students’
strategies?
• What does your description indicate
about his or her understanding of divisionabout his or her understanding of division
and/or multiplication

Third Grade
Represent and solve problems involving multiplication
and division.
3.OA.2 Interpret whole-number quotients of whole numbers,3.OA.2 Interpret 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 intoa 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.

Arrays in third grade helps students toArrays in third grade helps students to
make the connect with multiplication and
divisiondivision

Arrays in third grade making that connect to
multiplication and division
Repeated division with place value blocksRepeated division with place value blocks
Max the money loves bananas. Molly, hisy y,
trainer, has 24 bananas. If she gives Max
4 each day, how many days will they y y
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 the monkey loves bananas. Molly,a t e o ey o es ba a as o y,
gives Max 4 bananas each day, howg y,
H ld d ib t d t ’• How would you describe students’
strategies?
• What does your description indicate
about his or her understanding of divisionabout his or her understanding of division
and/or multiplication

How have you seen the
principals of UDLprincipals of UDL
demonstrated?
• Share your ideas on Wallwisher

Which type of DivisionWhich type of Division
Partitive or Quotitive?
Mrs. Campbell is arranging transportation for a
class trip She plans to drive and some parentsclass trip. She plans to 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 Campbell need forcar. How many cars will Mrs. Campbell need for
the trip?
Video Clip

Mrs. Campbell is arranging transportation forMrs. Campbell is arranging transportation for
a class trip. She plans to drive, and some
parents will too. Mrs. Campbell has 24
t d t i h l d h l tstudents in her class, and she plans to
assign 4 children to each car. How many
cars will Mrs Campbell need for the trip?cars will Mrs. Campbell need for the trip?
• How would you describe students’
t t i ?strategies?
• What does your description indicate abouty p
his or her understanding of division and/or
multiplication

Turn and TalkTurn 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 Mike’s mom uses 6 slicesA loaf of bread has 18 slices. 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. How many balls can Kevin buy?
• Katy is decorating goody bags for her birthday party.
She has 5 goody bags that she must decorate in theShe has 5 goody bags that she must decorate in the
next 35 minutes. How many minutes should she spend
on each bag?

Which examples do mostp
teachers provide for students in
their classroom?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.

Third Grade
Represent and solve problems involving multiplication and division.
3 OA 2 Interpret whole-number quotients of whole numbers e g interpret 56 ÷ 8 as the number of3.OA.2 Interpret 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.
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
lti li ti d di i i Ill t t d l i th l l ti bmultiplication and division. Illustrate and explain the calculation by
using equations, rectangular arrays, and/or area models.

Number in a GroupNumber in a Group
Using 4-digit by 1-digit
Mrs. Campbell’s class collected 3,468 cans
of food for the 3 food shelters If herof food for the 3 food shelters. If her
students divide the cans evenly among the
shelters how many cans of food would eachshelters how many cans of food would each
shelter get?
How might a number of groups problem look?

Algorithms for DivisionAlgorithms for Division
The long division algorithm is often difficult
f t d t t d d t dfor students to use and understand.
However, when teachers present an
bb i t d f t d t ’ d t diabbreviated form students’ understanding
is often sacrificed. Students demonstrate
l fi i i t th l ithless proficiency in carry out the algorithm
and make more errors.
NCCTN Developing Essential Understanding of
Multiplication and Division

Compounding the difficultly of divisionCompounding the difficultly of division
notation is the unfortunate phrase, “six goes
into twenty-four.” This phrase carries little
meaning about division especially inmeaning about division, especially in
connection with fair-sharing or partitioning
context. The “goes into” (or guzinta”)
i l i i l i d i d lterminology is simply engrained in adult
parlance and has not been in textbooks for
years. If you tend to use that phrase, it isyears. 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 class collected 3,468Mrs. Campbell s class collected 3,468
cans of food for the 3 food shelters. If her
students divide the cans evenly amongstudents divide the cans evenly among
the shelters how many cans of food
would each shelter get?ou d eac s e te get

Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
Standards for Mathematical Practice
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
othersothers.
4. Model with mathematics.
5 Use appropriate tools strategically5. Use appropriate tools strategically
6. Attend to precision.
7 L k f d k f t t7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

Third Grade
Represent and solve problems involving multiplication and division.
3 OA 2 Interpret whole-number quotients of whole numbers e g interpret 56 ÷ 8 as the number of3.OA.2 Interpret 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.
Fourth Grade
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,between multiplication and division. Illustrate and explain the calculation by using equations,
rectangular arrays, and/or area models.
Fifth Grade
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5 NBT 6 Find hole n mber q otients of hole n mbers ith p to fo r digit di idends and t o digit5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-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.

• Picture of division place value blocks

ResearchResearch
The national Council of Teachers of
M th ti d th t t d t h ldMathematics recommends that students should
“develop a stronger understanding of various
meanings of multiplication and divisionmeanings of multiplication and division,
encounter a wide range of representations and
problems situations that embody them, learnp y ,
about the properties of these operations, and
gradually develop fluency in solving
multiplication and division problems.”
(NCTM 2000, 149)( , )

Educational Approach with 3 Primary
P i i lPrinciples

R i f Y IdReview of Your Ideas
• How did you see the Three Principles of
UDL demonstrated in the math lesson?
Share your ideas from Wallwisher• Share your ideas from Wallwisher

Discussion
• What are the benefits of analyzing the
curriculum for strengths and weaknessesg
rather than focusing on the student’s
strengths and weaknesses? What are theg
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 ownas a deficit in their own
instruction, and endeavor
to develop the ability toto develop the ability to
discover a new method of
teaching.”
–Leo Tolstoy

Next Steps
• What are your next steps to integrate
UDL into your school environment?y
http://cast.org/

R fReferences
• 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
Mary Keel
Professional Development Consultant
252 725 2570919-807-3934
kitty.rutherford@dpi.nc.gov
252-725-2570
mary.keel@dpi.nc.gov
http://www wikicentral ncdpi wikispaces nethttp://www.wikicentral.ncdpi.wikispaces.net

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