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 and g 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/2011/11/30/norm-peterson-bought-the-house-next-door/

• Maintain Professionalism

• Participate Actively

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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 to p ydiscover 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 NetworkThe 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's y gstyle 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.

Universal Design for LearningUniversal Design for Learningprovides 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.

Universal Design for LearningUniversal Design for Learning

Purpose of UDL Curriculum

is not simply to help students master a specific body of knowledge or a specific p y g pset of skills, but to help them master learning itself—in short, to become expert g , plearners.

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 you g ythink are critical for t hi di i iteaching division.

ResearchSimply 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 between g pmultiplication 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 StandardsThird GradeOperations & Algebraic ThinkingRepresent 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 DivisionypPartitive 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 DivisionPartitive 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 MollyMax the monkey loves bananas. Molly his trainer, has 24 bananas. If she gives Max 4 bananas each day howgives Max 4 bananas each 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 divisionabout his or her understanding of division and/or multiplication

Common Core State StandardsThird GradeOperations & Algebraic ThinkingRepresent 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, his y y,trainer, has 24 bananas. If she gives Max

4 each day, how many days will the y y ybananas 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,his trainer, has 24 bananas. If she

gives Max 4 bananas each day, how g y,many days will the bananas last?

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 http://wallwisher.com/wall/hxqpnwkxac

Which type of DivisionWhich type of DivisionPartitive 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 about y phis 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 most pteachers 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.

Common Core State StandardsThird GradeOperations & Algebraic ThinkingRepresent 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 GradeNumber & Operations in Base Ten¹Use 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 DivisionThe 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 Practice1. 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.

How have you seen the principals of UDLprincipals of UDL demonstrated?


• Share your ideas on Wallwisher http://wallwisher.com/wall/hxqpnwkxac

Common Core State StandardsThird GradeOperations & Algebraic ThinkingRepresent 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 GradeNumber & Operations in Base Ten¹Use 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 GradeNumber & Operations in Base Ten¹Perform 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.

An Area Model for Division

• Picture of division place value blocks

Division with Decimals

0 80 ÷ 0 200.80 ÷ 0.20

0 30 ÷ 0 050.30 ÷ 0.05

ResearchResearchThe 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, learn p 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?


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Discussion

• What are the benefits of analyzing the curriculum for strengths and weaknesses grather than focusing on the student’s strengths and weaknesses? What are the gchallenges 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

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 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 RutherfordElementary Mathematics Consultant919 807 3934

Mary KeelProfessional Development Consultant252 725 2570919-807-3934

[email protected]@dpi.nc.gov

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