1. 4.2:EARLY NUMERACY AND BASIC FACTS Mathematical Thinking for Instruction K-3

2. Counting and Derived Facts Strategies Counting strategies typically involve demonstrating: 1. Quotity (holding a number) 2. The ability to ignore the time sequence (order) of a problem 3. Simultaneous double-counting Derived Facts strategies involve using what is known to solve what is unknown (Carpenter et al., 1999) June 16 2 © CDMT

3. Number Sense Number sense is…. “…good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.” -Howden, 1989

4. Counting The meaning for the counting sequence is the key conceptual idea Composed of two different skills a. Producing the counting words in order: “One, two, three, four…” b. Connecting the counting words with one-one correspondence to the objects counted. Counting with a purpose---‘chanting is not counting’

5. 5 Counting Principles One-one Correspondence: every item in a set must be given a unique ‘tag’ or ‘label’ (e.g. number name). Each item can only be one number name. Stable Order: each ‘tag’ or ‘label’ must be drawn from stably ordered list (e.g. number names). Cardinal Principle (or Cardinal Rule): the last number name used in a count has a special status; it represents the cardinal value of the set (e.g. ‘how many). Item-irrelevance: there are no restrictions on the items that can be counted. Order-irrelevance: the order in which the items are counted does not change the amount in the set. Gelman and Meck, 1986

6. Assessing the 5 Counting Principles Counting objects to solve problems (One-one) Watching someone else count incorrectly and responding (One-one) “1..2,3..4...5..6” “Can you start counting at N?” (OI, Card.) “How many are in the set? (OI, Card.) “What about now? N2134515234 Gelman & Meck, 1986

7. Write down the answers on your paper: How many tens are in 53? How many tens are in 1,037? How many tenths are in 238.7?

8. Place Value Concepts Packaging and Filling…with ‘left-overs’ Grouping beyond and within 10’s and 1’s Building a system of 10’s and 1’s There is a bag of toy car wheels. In the bag, there are enough wheels for 6 toy cars with 2 wheels left over. How many wheels are in the bag? Ross, 2002

9. Base-10 Problems ©CDMT 2008 Mallory has 7 bags of books with 10 books in each bag. She also has 3 books that aren’t in bags. How many books does Mallory have altogether? Steve collected 58 lucky stamps. Andy will trade him 10 lucky stamps for a Star Wars sticker. How many Star Wars stickers can Steve get with his 58 lucky stamps? A seed company sends seeds in envelopes containing an index card with 2 groups of 5 seeds taped to the card. How many envelopes will the company send for an order of 28 seeds?

10. Base-10 Problems ©CDMT 2008 How much is 3 tens and 5 ones? Kenzie has 72 gumdrops to share with the kids in her playgroup. Including Kenzie, there are 10 kids in the playgroup. If she doesn’t want to break or cut any gumdrops, how many gumdrops will each kid get? How many gumdrops will be left over?

11. Base-10 Problems ©CDMT 2008 How many tens are in 57? How many tens are in 243? Kevin has 243 pennies. It costs 10 cents to feed the goats. If he doesn’t get any more money, and doesn’t spend his money on anything else, how many times can he go and feed the goats this summer?

12. Basic Addition and Subtraction Facts Doubles/Reverse Doubles Doubles/Rev. Doubles+1/-1 Doubles /Rev. Doubles +2/-2 Making 10 (sometimes passing or bridging 10) 5 + 56 + 79 + 118 + 5 10 – 515 - 814 - 923 - 8 Possible basic fact framework for teachers:

13. Fluency and Flexibility Fluency- efficient and correct Flexibility- multiple solution strategies determined by the problem Fluency is the by-product of flexibility. Assessing fluency by occasionally using timed tests is acceptable. Using timed tests as an instructional tool to build fluency is ineffective, inefficient, and damaging to student learning. Baroody, 1985; Brownell, 1935; Dawson & Ruddell, 1955; Fuson, 1992; Henry & Brown, 2008

14. Representational Progression: How Children Understand Formal Symbolism Enactive: tangible, experiential Iconic: direct representations of ‘reality’ Symbolic: formal signs and symbols with culturally mediated meanings (sometimes referred to as Concrete, Representational, and Abstract) To address this progression: Manipulatives (if needed) Drawings to represent manipulatives (or processes) Label drawings with numbers Number sentences from the labels Bruner, 1966

15. Representational Progression Example: From Drawings to a Number Sentence Aisha counted 8 cars in the parking lot. 5 cars drove away. How many cars are left? 1.2. 3.4.