1. CHAPTER 8 Developing Early Number Concepts and Number Sense
Elementary and Middle School Mathematics Teaching Developmentally
Ninth Edition
Van de Walle, Karp and Bay-Williams
Developed by E. Todd Brown /Professor Emeritus University of Louisville

2. Big Ideas Counting tells how many things are in a set.
Numbers relate through comparisons of quantities.
Number concepts are intimately tied to operations with numbers based on situations in the world around us.
Number sense means that you can think about different-sized quantities and use numbers and relationships in multiple ways to estimate and solve problems.

3. Research-based recommendations for high-quality learning activities in the first 6 years of life
Enhance children’s natural interest in mathematics
Build on children’s experience and knowledge
Base mathematics curriculum and teaching practices on solid mathematical knowledge and child development
Use formal and informal experiences to strengthen children’s problem-solving and reasoning processes
Provide opportunities for children to explain their thinking as they interact with mathematical ideas
Assess children’s mathematical knowledge, skills, and strategies through observation and other informal practices.

4. Four Categories of Early Number Relationships
Spatial Relationships One More/Two More/One Less/Two Less
Anchors to 5 and 10 Part-Part-Whole

5. The Number Core: Quantity, Counting and Knowing How Many
Quantity and the ability to subitize
When you look at an amount of objects, sometimes you are able to just “see” how many are there.
Activity 8.1
Materials- counters, paper plates, dot plates
Directions- hold up dot plate
“Make the pattern you saw on this
plate with your counters”
Show plate again as a self-check
* Include zero and what it means when there are no dots

6. Early Counting
Counting the sequence of number words is rote.
The meaning attached to the counting is the key conceptual idea on which all other number concepts are developed.
Research-based learning trajectories provide a path for instructional direction
Verbal counting two separate skills
Standard list of counting words
Connect the sequence in a one-to one correspondence with objects
Children will learn how to count before they understand that the last count word indicates the amount of the set or the set’s cardinality

7. Meaning Attached to Counting of Objects
In counting, each number is one more than the previous number

8. Diagnostic Assessment
Formative Assessment
Show a card with five to nine dots in a row
How many dots are on the card?
Children who respond with the correct number have made the connection to the cardinality principle.
Please get the same number of counters as there are dots on the card. (indicators next slide)

9. Sequence of Indicators to Watch For
These are in order from a child who does not attach meaning to the count to one who is using counting as a tool:
Does the child not count, but instead take out counters and make a similar pattern?
Will the child recount?
Does the child place the counters in a one-to one correspondence with the dots?
Does the child just count the dots and retrieve the correct number of counters?
Can the child show that there are the same number of counters as dots?

10. Numeral Writing and Recognition
Reading and writing numerals is similar to teaching them to read and write letters of the alphabet. Engaging ways: Make numerals with clay Trace in shaving cream Write them on the interactive whiteboard Calculator keypad

11. Relations Core: More Than, Less Than, and Equal To
Children need to construct sets using counters and make comparisons or choices between two or more sets.
Activity 8.10
Materials- picture of sets of objects of 4 to 12, set of counters, word cards labeled More, Less, Same
Directions-Next to each card have children make three collections

12. Number Relationships 1-10
One more and two more, one and two less
Anchors or benchmarks of 5 and 10
Part-part-whole relationships
Counting is the principal tool children will use as they construct these relationships.

13. Building Relationships
Activity 8.13 Materials- six dot cards, counters Directions- Ask children to display a set of counters that is two more than the set shown.
Activity 8.17 Materials- little ten frame cards Directions- Flash the ten frame and ask “how many dots are shown?” Variations How many spaces instead of dots? Six and what number equal 10?

14. Part-Part-Whole Basic Ingredients
Compose
Build designated quantities in two or more parts
Decompose
Start with a full amount and separate into two or more parts
Children need to read and write the combinations for a clear connection

15. Number Relationships 10-20
A set of ten plays a major role in the the understanding of numbers
Three-level progression of children’s understanding of ten:
Child understands ten as ten ones and does not see ten as a unit.
Child understands ten as a unit but relies on materials to complete tasks involving ten.
A facile concept of ten- solves tasks without using materials

16. Ten and Some More Activity 8.26
Children do not see a numeric pattern in the numbers 10-20
Core Content for State Standards suggest children think of 10 as a bundle of ten ones- called a ten
Think of the teens as a ten and some more ones
Activity 8.26
Materials – two part mat, counters and a story
Directions- ask students to complete the ten frame and count out 7 counters on the other side
Have them count altogether
Chorus the combination- ten and seven equal
seventeen, turn mat around- seven and ten
equal seventeen

17. Number Sense in Their World
Estimation and Measurement
Cut out a large footprint 18 inches long. Ask children to tell you about how many will it take to measure the rug in our classroom?
More or less than_____?
Closer to ___ or to _____?
About ______?
Connecting numbers to real situations- Activity 8.28 and Suggest units to use with a number nine ____. Also give a number and unit and figure out when that would be reasonable-15 feet.

18. Data collection and analysis
Encourage children to generate ideas for what data to gather.
Use the graphed data to compare number relationships rather than only referencing more and less.
More students brought apples for snack than bananas.
Three fewer people brought oranges than brought apples.