K-2 Standards and Support Training

Office of Standards and Learning

Mary L. Ruzgamruzga@ed.sc.gov Summer 2015

Introductions• Introduce yourself to your table mates

• Show of hands – How many• Pre-K, K, 1st, 2nd, etc., Principal/Assistant,

District Leader, other?• have participated in Numeracy Leader

training?• have read the new standards?• are comfortable with the new standards?

What will we do today:

1. Quick overview of format of the standards

2. Review the standards by engaging in activities which explain and demonstrate the need for possible new pedagogical techniques

3. Review and explore the K-2 portion of the Mathematics Support Document

Accessing the Standards and Resourceswww.ed.sc.gov

BASIC COURTESIES/REMINDERS

Please turn off phones (emergency vibrate)

Please refrain from texting, emailing, etc.

No formal break – take care of personal needs

Logistics (lunch, restrooms)

Participate and respect those around you

Content Standards Process Standards

Portrait of SC Graduate

Process Standards

Mathematical Process Standards

1. Make sense of problems and persevere in solving them.

2. Reason both contextually and abstractly.3. Use critical thinking skills to justify

mathematical reasoning and critique the reasoning of others.

4. Connect mathematical ideas and real-world situations through modeling.

Source: South Carolina College- and Career-Ready Standards for Mathematics 2015

Mathematical Process Standards

5. Use a variety of mathematical tools effectively and strategically.

6. Communicate mathematically and approach mathematical situations with precision.

7. Identify and utilize structure and patterns.

Source: South Carolina College- and Career-Ready Standards for Mathematics 2015

Content Standards

Format of Mathematics Content Standards

K – 8 Grade Level Content Standards

Source: South Carolina College- and Career-Ready Standards for Mathematics 2015

Format of Mathematics Content Standards

K – 8 Grade Level Content Standards

Source: South Carolina College- and Career-Ready Standards for Mathematics 2015

In grades K – 8:GradeLevel.KeyConcept.Standard

Number (e.g., K.NS.1) or, if applicable,GradeLevel.KeyConcept.Standard

NumberStandardLetter (e.g., K.NS.4a)

Format of Math Content Standards

Source: South Carolina College- and Career-Ready Standards for Mathematics 2015

Termsincluding - references content that must

be mastered (See K.G.1)

e.g. - references possible illustrative examples. (See 1.NSBT.2c)

i.e. - references the only examples or terms that should be used. (See 2.MDA.3)

Source: South Carolina College- and Career-Ready Standards for Mathematics 2015

TermsFluently and fluency describe a student’s ability

to compute with accuracy, flexibility, and efficiency (Kilpatrick, Swafford, & Findell, 2001).

Real-world refers to authentic contexts through which students engage in mathematics and should serve as a stepping-stone for thinking about important mathematical concepts.

Source: South Carolina College- and Career-Ready Standards for Mathematics 2015

Digging Into the K-2 Standards for Mathematics

K 1st Grade 2nd Grade 3rd Grade 4th Grade 5th GradeNumber Sense

Number Sense and Base Ten

Number Sense and Base Ten

Number Sense and Base Ten

Number Sense and Base Ten

Number Sense and Base Ten

Number Sense and Base Ten

Number Sense - Fractions

Number Sense and Operations - Fractions

Number Sense and Operations - Fractions

Algebraic Thinking and Operations

Algebraic Thinking and Operations

Algebraic Thinking and Operations

Algebraic Thinking and Operations

Algebraic Thinking and Operations

Algebraic Thinking and Operations

Geometry Geometry Geometry Geometry Geometry Geometry

Measurement and Data Analysis

Measurement and Data Analysis

Measurement and Data Analysis

Measurement and Data Analysis

Measurement and Data Analysis

Measurement and Data Analysis

Progression of Key Concepts

Digging Into the Standards

Look at the standards for Grades K-1. Discuss with your tablemates:

• If you count the standards and the a,b,c, etc. which Key Concept is the largest in each grade?

• Why do you think that is so?

Digging Into the Standards

Look at the standards for Grade 2. Discuss with your tablemates:

• If you count the standards and the a,b,c, etc. which Key Concepts are the largest?

• Why do you think that is so and what is the relationship between those concepts?

Digging Into the Standards

If grades K-1 have the largest amount of standards devoted to number sense where should the bulk of our teaching time for mathematics be spent?

Two types of Number Sense:• Pre-operational• Operational

Pre-operational mathematical development is one of the most important predictors to future mathematical success.

Meaning Attached to Counting. . .

“Generally, children at midyear in kindergarten should have a fair understanding of counting, but children must construct this idea. It cannot be forced. Only the counting sequence is a rote process. The meaning attached to counting is the key conceptual idea on which all other number concepts are developed.” (Emphasis added)

John A. Van de Walle

Digging Into the Standards

What is meant by Pre-operational Number Sense?

Compare: K.NS.1 and K.NS.2 to

K.NS.3, K.NS.4 and K.NS.5

Discuss with your tablemates:• How do the expectations in the first two standards

differ from the expectations of the second three standards?

Rote vs meaningful counting

Digging Into the Standards. . .

Pre-Operational Number Sense (meaningful counting)

Look at:K.NS.4a (cardinality)

• Counting Strategy• 1, 2, 3, 4, 5 How many?• 1, 2, 3, 4, 5 How many?

K.NS.4b (conservation of number)

• Arrangement Strategy• Count, spread out and re-count

K.NS.4c (increasing/decreasing magnitude)

• One/two more/less K.NS.5 (one-to-one correspondence)

Digging Into the Standards

Comparison of Quantities - Number Relationships

Discuss with your tablemates:• How do K.NS.7 and K.NS.8 differ from

each other and how do both relate to K.NS.4c?

All three deal with number relationships – how does one quantity relate to the other (increasing/decreasing magnitude).

Digging Into the Standards

Comparison of QuantitiesNumber Relationships versus Operations

Think about:If comparisons are number relationships – how one quantity relates to the other – then is that counting or is it operations and as a teacher does it matter?

Re-read K.NS.4c

Let’s explore through an activity.

Is it counting or is it operations? Does it matter?

K – Play “More/Less Than” with five frameNeed: 5 frame; baggie of pink cards and paper

clip; spinner on “Spinning More or Less”, “One More Than Recording Sheet”; and follow directions on recording sheet

1st and 2nd Grade – Play “Spinning More/Less” (Bingo)Need: one die, paper clip from baggie of pink

cards; several two-color counters (one color for you; other for partner); “Spinning More/Less”, and follow directions on spinning sheet

Is it counting or is it operations? Does it matter?

Talk with your tablemates:• Which game would most likely support a

counting solution strategy and which one an operational solution strategy?

Read 1.ATO.5Talk with your tablemates:• How does counting relate to addition and

subtraction?

Digging Into the Standards

Read the following Standards:• 1.NSBT.5• 2.NSBT.2• 2.NSBT.8

Then discuss with your tablemates:• How does the K concept of counting “one

more/less” relate to the above standards?• If students are struggling with the above

standards, what might you need to consider?

Extending Number Relationships to Multiples

Understanding the increasing/decreasing principle (number relationships) is necessary in order to count and add by multiples.

One ten and one more ten is?

In addition to number relationships, students also need to have developed the concept of subitizing in order to count on or “add on”.

Digging Into the Standards

Visualization“Sight Words/Numbers/Quantities”

Look at KNS.6 • Perceptual Subitizing• Conceptual Subitizing

Beginning of counting on . . . (Add)

• Perceptual Subitizing – instantly seeing how many in a small set of 2-3 objects; recognizing a # without operational knowledge

• Conceptual Subitizing – similar to the ability to recognize a # without operational knowledge but with larger sets and recognizing the number as both a whole and as its composite parts

Digging Into the Standards

Visualization“Sight Words/Numbers/Quantities”

Look at KNS.6 • Perceptual Subitizing• Conceptual Subitizing

Beginning of counting on –

Play “What Number Do You See?”

Digging Into the Standards

Read:K.ATO.4, 1.NSBT.2 - all parts 1.ATO.3, & 6, 2.NSBT.1 - all parts 2.ATO.2

Then discuss with your tablemates:• How does the ability to perceptually and

conceptually subitize relate to the above?

• If a student is struggling with the concepts in the above standards, what might you check for?

How might the concept of subitizing be developed in relation to operations? (Play How Many are Hiding? )

Subitizing/Visualization Needed for Counting On and as Foundation for Operations. . .

Play – “How Many are Hiding?” K – use 5 cubes; 1st and 2nd grade - use 10 cubes

Need: Paper plate, cm cubes1. Count out cubes; Partner A grab some of the cubes and hide under plate; ask Partner B –

• How many are hiding?• How do you know?

2. Partner B states:• I know ___ are hiding because ____ and ___

more equal 5/10.

3. Count to verify Partner B’s response4. Exchange Roles and Repeat

Subitizing/Visualization Needed for Counting On and as Foundation for Operations. . .

“How Many are Hiding?”

Handouts• Continuum of Understanding• Stages of Counting by Multiples

Read K.ATO.2 1.ATO.1 2.ATO.1Addition is a Joining/Part-Part Whole action Parts of the Whole, Separation, Comparison

Digging Into the Standards

Operations:K.ATO.1 and 2

Digging Into the Standards – “Start Unknown”

1.ATO.4 Understand subtraction as an unknown addend problem.

Sandra had some pennies. George gave her 4 more. Now Sandra has 12 pennies. How many pennies did Sandra have to begin with? (Join: Initial Unknown/ Add to: Start Unknown)

_____ + 4 = 12 Semantic Form of the equation12 – 4 = _____ Computational Form

What might we do in our classrooms to help students come to see the two forms as equivalent?

Once you know students can subitize and count on, focus on the operations – Subtraction as Think Addition and vice versa .

How would the game, “How Many are Hiding” relate to 1.ATO.4 if students were required to write the number sentences?

How many are hiding and how do you know?• I know 7 are hiding because 10 – 3 (amount showing) = 7 (amount hiding)

You are right, 7 are hiding because 3 + 7 = 10

3. Count to verify number sentences4. Exchange Roles and Repeat5. Results can re recorded in a variety of forms.

18

7

38

11

24

35Number Bonds

Funnel Machine

24

4

10

10

38

31

7

Number Machine

Another design for Number Bonds

Digging Into the Standards

The tendency in the US is to have children solve a lot of problems in a class period and focus on the correct answer rather than focusing in-depth on one or two well-chosen problems at a time, the relationship between the structures of the problems and the variety of approaches to the solutions. John A. Van De Walle

Digging Into the Standards

1.ATO.3 vs 2.ATO.4 Read the above two standards then discuss with your tablemates:How do the student expectations in these two standards differ?

• Equal Groups • concrete objects

• Size of each group (4)• Lesson 2 Handout

• Number of groups (3)

4 + 4 + 4 = 12Handout x 2

Important Considerations at each Grade K-2• Rote Counting• Meaningful Counting

Cardinality (last number spoken) Conservation of Number (arrangement) Increasing/decreasing Magnitude Principle

(one/two more/less) (number relationships) One-to-One Correspondence Subitizing (visualization) Comparisons (quantity relationships) Check for student solution strategies in

order to determine: Is it counting or is it operations? This determines how to move the student forward in understanding.

Support Document

Support Document Overview

Mathematics Support Document Overview

“Overview”

http://ed.sc.gov/agency/ccr/Standards-Learning/Mathematics.cfm

o Mathematics Edmo Group

Mathematics Support Document

Review Process: Work by grade levels – YOUR DECISONo Move into Grade Alike Groupso Open browser

http://maryruzga.weebly.com Not hyperlinked at this time

Grade Levels Review/explore Share out one activity/lesson/resource

SCDE Mathematics Team: https://edmo.do/j/snmcix

1. Create an account. (If you have an account, log in.)

2. Enter URL above.3. Different Grade Band “Folders”

Joining Mathematics Edmodo Group

Office of Standards and Learning: https://edmo.do/j/fu7i2y

1. Create an account. (If you have an account, log in.)

2. Enter URL above.

Joining Office Edmodo Group

Mathematics Support Document

Feedback and Certificate of Participation

https://scde.formstack.com/forms/summer_plo_evaluation_form_copy_1_copy

See Yellow Handout for link or the QR code

Geometry Handout

My Contact Information

Mary L. Ruzgamruzga@ed.sc.gov

Thank you for spending time with me today.