Developed byCarmen Davis, Hoa Ly, Ryan Ratajski

& Johari SykesCamden City School District

Understanding Numerical Operations

in Preschool

Standard 4.2 Children demonstrate an initial understanding of numerical

operations.

Match The Cards•Select a piece of paper from the bag provided on your table.• Try to find a member of the group with the matching halfof the phrase.•After matching the phrase, take this opportunity to talk to your partner about some things you know about numerical operations.

Today’s ObjectivesTo define numerical operations, giving teachers a

deeper understanding of what the math content means.

To describe how this standard translates into practice during small group times, large group times, and as a choice embedded in classroom centers.

To highlight math practices that can be used while teaching about numerical operations.

To accentuate that the quality of the teacher-child interactions is key to enhancing children’s outcomes as they develop an initial understanding of numerical operations.

2004: Math Expectation 1Children demonstrate an understanding of number and numerical operations

Preschool Teaching Practices•Make materials and books that promote number exploration accessible to the children.•Encourage children to compare numbers frequently through questions and graphing.•Integrate purposeful counting experiences throughout other learning opportunities.•Encourage and support individual attempts to learn to count numbers. Encourage counting to 30+.

2004: Math Expectation 1Children demonstrate an understanding of number and numerical operations

Preschool Teaching Practices (cont’d.)•Provide children opportunities to use estimation skills during daily activities by asking interesting and relevant questions.•Model addition for children by using counting to combine numbers.•Model subtraction for children by using counting to separate numbers.•Foster one-to-one correspondence throughout the day.

2004: Math Expectation 1Children demonstrate an understanding of number and numerical operations

Preschool Learning Outcomes1.1 Demonstrates understanding of one-to-one correspondence.1.2 Spontaneously counts for own purposes.1.3 Learns to say the counting numbers.1.4 Discriminates numbers from other symbols in the environment.1.5 Recognizes and names some written numerals.

2004: Math Expectation 1Children demonstrate an understanding of number and numerical operations

Preschool Learning Outcomes (cont’d.)1.6 Compares numbers in different.1.7 Uses estimation as a method for approximating an appropriate amount.1.8 Adds two groups of concrete objects by counting the total.1.9 Subtracts one group of concrete objects from another by taking some away and then counting the remainder.

What does “numerical operations” mean for our

preschoolers?

Numerical OperationsPreschool Learning Outcomes

4.2.1Represent addition and subtraction by

manipulating up to 5 objects:

(a) putting together and adding to (e.g., “3 blue pegs, 2 yellow pegs, 5 pegs altogether.”); and

(b) taking apart and taking from (“I have four carrot sticks. I’m eating one. Now I have 3.”).

4.2.2Begin to represent simple word problem data in

pictures and drawings.

Number Sense + Numerical Operations

= Number & Operations

• Counting (1, 2, 3…one-to-one correspondence)

• Quality (sense of number; three fingers = 3; subitizing)

• Order (sequence – first, second, third…)

• Numerals (seeing numbers and what they represent)

Number Sense + Numerical Operations

= Number & Operations (cont.)

• Comparisons (more/less; more/fewer)

• Combining operations (adding)

• Separating operations (subtracting)

• Sharing operations (dividing)

• Set-Making operations (multiplying)

Numerical OperationsPreschool Learning Outcomes

4.2.1Represent addition and subtraction by manipulating up to

5 objects:

(a)putting together and adding to (e.g., “3 blue pegs, 2 yellow pegs, 5 pegs altogether.”); and

(b)taking apart and taking from (“I have four carrot sticks. I’m eating one. Now I have 3.”).

4.2.2Begin to represent simple word problem data in pictures

and drawings.

Numerical OperationsWhat Preschoolers Can and Cannot

Do

Preschoolers typically:

Believe that two sets of objects have the same number if the objects are close to one another.

Solve subtraction problems before they can solve addition problems.

Think about small numbers without having the physical objects.

Preschoolers typically do not:

Understand mathematical words in abstract thinking without the use of real objects present.

Connect the idea on large quantity schema.

Easily keep track of what they have counted or not counted.

Teaching Practices and Classroom Environments that

Make the Difference

Preschool teachers will:Model addition for children by using counting to

combine numbers (e.g., “Maria has two blocks and Justin has three. There are five blocks altogether: 1, 2, 3, 4, 5.”).

T: How many green stripes do you have?

C: (counts) 1, 2.T: How many yellow stripes do you

have?C: (counts) 1, 2, and one glitter one,

too.T: Good. So how many do you have

in total? That means altogether.C: (counts) 1, 2, 3, 4, 5. I got five

stipers!

C: I hit all them pins down, but I can pick them back up.

T: Why would you do that?

C: Because if I pick one up, then it’s one and then if I pick up this one, it’s two of them. (counts) 1, 2, 3, 4, 5, 6, 7, 8. It’s eight of them still down.

T: This is a number sentence. How many children are here?

C: (counts) 1, 2, 3. 3.T: Plus how many children?C: One.T: Equals…C: (counts) 1, 2, 3, 4. Four kids. It’s

four.T: Can you find that card in your

hand?C: Yes. (selects the number 5 card

and holds it up)

Video #1

Scaffolding Activity

What did you see and hear in this clip? Use the checklist to guide a discussion at your table.How does this teacher’s activity support Standard 4.2.1:

Represent addition and subtraction by manipulating up to 5 objects?

Then, as a group, use the 5 objects in the brown bag to modify the activity seen in the clip or to develop a new activity. Think about children in the group who might be at earlier stages of understanding and/or children with more advanced thinking.

To debrief, we’ll have one person from each group sum up the group’s discussion and another person from each group describe the activity developed by the group.

Materials to Use

Linking cubes

Collections of natural objects Counting bears

Play people

Preschool teachers will:Develop addition and subtraction stories with small groups of children using story mats and flannel board scenes with small quantities of objects and pictures/drawings.

Preschool teachers will:Use fingers, chalk, wipe-off markers and/or whiteboard technology to tell and draw addition and subtraction stories with small groups of children.

Preschool teachers will:Provide writing materials and/or handheld devices with appropriate applications in classroom centers so that children can choose to view, solve, and create addition and subtraction stories.

Figure 1 Figure 2

Let’s try it!

Represent this math problem With picturesWith objectsBy acting it out

Video #2

Assessing Your Current Practice

From Preschool to KindergartenDo you know what the Common Core asks of

children in kindergarten?

New Jersey Preschool Teaching and Learning Mathematics Standards Aligned to the Common Core Mathematics Standards for Kindergarten – DRAFT

Making It Happen

ReferencesThe Creative Curriculum for Preschool

- Volume 4 Mathematics by Juanita V. Copley, Candy Jones, and Judith Dighe

The Young Child and Mathematics by Juanita V. Copley

Resources for Further Usehttp://www.naeyc.org/positionstatements/

mathematicsNAEYC/NCTM position statement on Early Childhood Mathematics: Promoting Good Beginnings

Teaching and Learning Math: The Learning Trajectories Approach by Douglas H. Clements and Julie A. Sarama

http://www.youtube.com/watch?v=wV85P07K1mUThe Building Blocks of Math: Lessons from Research, featuring Dr. Doug Clements