Developing Number SenseAmy LeHew – Elementary Math

August 2011RTI Kathy Richardson

Assessment Pilot

Memorized knowledge is knowledge that can be forgotten. Internalized knowledge can’t be forgotten

because it is a part of the way we see the world.

Welcome Teachers!

Today’s Agenda 

8:30-11:30

Understand how children come to learn numberDiscuss challenges to learning and methods to meet their needsExplore Assessment Series

11:30-12:30

11:45-1:00 Lunch

12:30 – 1:30 computer lab

FamilyTeaching (Iowa)MathWater

Amy LeHew FamilyTeaching (Iowa)MathOcean

Let’s Start with some MATH

Welcome to Numeria my young Numeriticians!

Today is only our first day, so let’s count!

XiknXNunXiXkXn

NumeriaLet’s try this with manipulatives. Grab some cubes and lets try again…

X, i, k, n, Xnun, Xi, Xk, Xn

Let’s Start with some MATH

X, i, k, n, Xnun, Xi, Xk, Xn

Ok, how would we write the next one?

X, i, k, n, Xnun, Xi, Xk, Xn

Hmmm, you clearly need another support.

Let’s try filling our X-Frame

Numeria

X, i, k, n, Xnun, Xi, Xk, XnLet’s try some simple addition and subtraction

x + k=n – iXi + ? = xn

Place Value

What base were we working in?

At your table, discuss the importance of understanding base-ten structure.

Where do students struggle? What can we do to help them?

Expectations 

Students learn to count in kindergarten.Students leave first grade knowing some combinations to ten.By the end of 3rd grade, students are expected to add 3-digit numbers.Many students are unable to make this transition because they do not have a solid understanding of the structure of number.

So how do students develop powerful number sense?

Counting videos

Making Ten

How many? How many more to make ten?

What does a student need to know before they can engage in this type of activity?

What happens when students struggle?

The Curriculum

Developing Number Concepts by Kathy RichardsonDesigned to help children internalize foundational math concepts Utilize a variety of learning activities. Provide ongoing assessment and activities designed to meet individual student needs.

The Curriculum

Assessments:Interview-styleAdministered onlineTake about 5 minutes (each)Pinpoints what students

understand about numberBased on Kathy Richardson’s Critical Learning Phases

The Curriculum

Lessons:

Lessons to useDesigned to help children internalize foundational math concepts Easily integrated into Math Workshop timeBased on student needs

The Curriculum

Amcanywhere.com

Record as you assessProvide detailed reports for each student and class

Where should we start?Assessment # 5: Number Combinations

ByKathy Richardson

Assessment #5Combination Trains

Overview & Description of

Strategies

Learning Number Combinations

• Children need to see the basic facts as a set of interrelated concepts.

• Children need to be able to look for relationships

between the facts they know and other larger, more complex numbers or problems.

• Emphasis needs to be on learning number composition and decomposition and number relationships – not just on getting the right answers.

What are we trying to determine with this assessment?

To determine what number combinations the student knows and to find out if they can use the answer to a combination they know to figure out one they don't know.

Does student know the parts of numbers to 10?

Can student use efficient strategies to solve problems to 20.

What will my students be asked to do during the Combination Train

assessment?

• Students will be presented with connecting cube trains of different lengths – they will be asked to add a variety of number combinations.

• Will assess their fluency with numbers to 6, to 10, and to 20.

Use “Linking Assessment to Instruction” guides for instructional support from Developing Number Concepts

Select Downloads

Select Linking Assessment

ByKathy Richardson

Assessment #6Hiding Assessment

Overview & Description of

Strategies

Learning to Decompose Numbers

• To subtract children need to know the parts of numbers and see the relationship between composition and decomposition.

• Children must recognize that one

number is contained within another number.

• Children must understand that the number stays the same even when it is broken apart and recombined in various ways.

What are we trying to determine with this

assessment? Does the student… -know parts of numbers to 10 quickly, without counting to figure them out.

Can the student…-use what they know about parts of numbers to solve subtraction problems.

The Hiding Assessment: Video

Show video 24:10

Look at student behaviors. What does it look like when they…

Don’t know?Are confident?

25:11 Kindergartener with 5 cubes26:36 First Grader: 5 and 628:42 Second Grader; 6 and 10

What Strategies do Students Use?Knows Quickly: Does not hesitate

or count to figure out.Related Combinations: Uses what they already know to figure out what they don't know. Ex: I see 2. 4 and 2 is 6 so 4 are hiding.

Counts On or back: Starts with what they see and counts on or starts with the whole number and counts back for each counter they see. "I see 4...So, 5...6 are hiding. Two are hiding."Counts All: Uses fingers or

visualizes the whole number.

Counts On or back: Starts with what they see and counts on or starts with the whole number and counts back for each counter they see. "I see 4...So, 5...6 are hiding. Two are hiding."

I’ve Assessed, Now What?

Assessment bookPage 52 – 54

2:2-5Book 2Chapter 2Activity 5

Look in DevelopingBook 2 – page v

ByKathy Richardson

Assessment #6Hiding Assessment

What questions do you have?

ByKathy Richardson

Should we go back? (to assessment #5)

Or Forward?(to assessment #6)

ByKathy Richardson

Assessment #5Combination Trains

Overview & Description of

Strategies

Learning Number Combinations

• Children need to see the basic facts as a set of interrelated concepts.

• Children need to be able to look for relationships

between the facts they know and other larger, more complex numbers or problems.

• Emphasis needs to be on learning number composition and decomposition and number relationships – not just on getting the right answers.

What are we trying to determine with this assessment?

To determine what number combinations the student knows and to find out if they can use the answer to a combination they know to figure out one they don't know.

Does student know the parts of numbers to 10?

Can student use efficient strategies to solve problems to 20.

What will my students be asked to do during the Combination Train

assessment?

• Students will be presented with connecting cube trains of different lengths – they will be asked to add a variety of number combinations.

• Will assess their fluency with numbers to 6, to 10, and to 20.

Trains

How is this different from the Hiding Assessment? .

ByKathy Richardson

Assessment #7Ten Frames

Overview & Description of

Strategies

Learning about Numbers as One Ten

and Some More• Understanding that numbers are made up of

“ten and some ones” is a foundational skill students must learn to work with larger numbers.

• To solve more challenging problems student must move beyond counting on strategies and be able to solve problems by using relationships and understanding the underlying structure of numbers to 20.

What are we trying to determine with this assessment?

To determine if the student can combine single digit numbers by reorganizing them into a 10 and leftovers.

To determine if the student can use their knowledge of the parts of numbers to 10 to subtract numbers up to 20.

What will my students be asked to do during the

Ten Frames Assessment?

Using ten frames, students answer a series of addition and subtraction questions.

Assesses whether students know parts of numbers and can break numbers apart to complete the ten to solve the problem.

To complete assessment, students solve “what if” questions without the support of the ten frame.

Overview of Ten Frames Assessment

Part 1 – Addition• Part 2 – Subtraction

Each part divided into four areas:Adding/Subtracting Ten and OnesKnows Parts of NumbersMaking a Ten and Adding/Subtracting OnesRecognizes Ten More

Note: Ten frames to use during assessment are available in download section.

Part 1, AdditionAdding Ones to a Ten (10 +9 and 6 +10):

Trying to assess if student can easily add the ones to the ten without counting on or

counting all. These strategies are self-explanatory.

If “N” assessment will end.

Student is presented with multiple problems and is asked to break apart the number to make ten (knows parts of numbers) and then asked how many altogether (making a ten and adding ones). The final question adds ten to the question (recognizes ten more).

• Use practice question 9 + 3 to be sure student understand procedure.

Part 1, Addition continued

Knows Parts of Numbers

Making a Ten and Adding Ones8 + 7, 7 + 6, 8 + 5

Recognizes Ten More18 + 5

Part 2 - Subtraction

• To determine if the student can decompose a teen number into a ten and leftovers.

• To determine if the student can subtract by breaking up a number in order to get to ten and then subtracting what is left from 10.

• Practice Problem: 12-3

ByKathy Richardson

Assessment #8Grouping Tens

Overview & Description of

Strategies

Learning about Numbers as Tens and Ones

Children need to learn that numbers to 100 are composed of groups of tens and ones.

Children must do more than label the digits in a number – they must understand that numbers are organized into groups of tens and ones.

Children must recognize that a ten is both one ten and ten ones. This level of thinking is difficult for young children.

What are we trying to determine with this assessment?

Do students understand that numbers to 100 are organized into groups of tens and

ones.

Can they add tens without counting.

Can they take away tens without counting.

Do they understand that counting by groups doesn’t change the total quantity?

What will my students be asked to do during the Grouping Tens assessment?

• Students will be asked to identify tens and ones when presented with number cards. Note: Number cards available under downloads of www.amcanywhere.com.

• Using counters, student will identify groups of ten and leftovers.

• Student will use what they know about tens to find out how many altogether.

• Students will be asked to add and take away tens without counters.

Assessment #8: Grouping Tens

• Video: 33:20• First Grader and Second Grader: 33:50 – 37:10• 40:12 Tad Second Grader •

ByKathy Richardson

Assessment #9Two-Digit Addition

& Subtraction

Overview & Description of

Strategies

Learning to Add & Subtract Two-Digit Numbers

• Students must understand the underlying math to solve two-digit addition & subtraction problems – otherwise they will simply be following a procedure.

• They must have a solid understanding of number relationships, number combinations to ten and numbers as tens and ones for two-digit addition & subtraction to have meaning for them.

• Children must have repeated practice combining, separating, and regrouping numbers.

What are we trying to determine with this assessment?

To determine if the student can use the concept of tens and ones to add two-digit numbers, by mentally breaking them apart and reorganizing them into a total number of tens and ones, when the problem is presented

1) using models2) with the model covered and 3) symbolically

What will my students be asked to do during the Two-Digit Addition &

Subtraction assessment?

Students will be asked to solve a series of two-digit addition and subtraction problems.

They will be asked to solve the problems by grouping/regrouping the numbers into tens and leftovers.

They will be asked to use models appropriately to

show their thinking, and they will be asked to solve the problems without models.

Overview of Two-Digit Addition & Subtraction

AssessmentPresent a variety of two-digit problems to student:

• Solves Problem with Model – showing trains of cubes• Solves Problem w/out Model – cover cubes with

paper• Solving Symbolic Problems – use equation cards

Assessment is capturing student’s answers, their strategies for solving the problems and their use of the models.

Equation Cards available in the Downloads section ofwww.amcanywhere.com

What are the choices for strategies during the Two-Digit Addition & Subtraction Assessment?

Knows parts to make/adds ten: Student easily forms tens; knows parts of numbers without having to figure anything out.

Counts to make/add ten: Student can form tens and leftovers, but not automatically and still needs to figure them out.

Visualizes written problem: Trying to solve problem using standard algorithm.

Counts all or on: Student still thinks of numbers as a collection of units (rather than tens & ones) and counts all or on to get answer.

Strategy Unknown: Prompts teacher to ask“How did you think about that?”

Assessment Results

Summarized at end of assessment as:

A – Ready to Apply

P – Needs Practice

I – Needs Instruction

Complete descriptions included in assessment guide.

Reflection/ TaskA. Complete the school information sheet (in the center

of the table).B. Individually, respond to the following questions on a

notecard.

1. What (if anything did you learn today) about number sense, student understanding and/or mathematics?

2. What else do you want to know?

3. What would you have LIKED to see/do/get from this session? (don’t say computer because we’re going there)