“… a well organised conceptual framework of number information that enables a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms" (Bobis, 1996).
The National Council of Teachers (USA, 1989) identified five components that characterize number sense:
operations involving numbers
referents for numbers and quantities
Researchers have linked good number sense with skills observed in students
proficient in the following mathematical activities:
computational estimation (for example; Bobis, 1991; Case & Sowder, 1990)
judging the relative magnitude of numbers (Sowder, 1988)
recognizing part-whole relationships and place value concepts (Fischer, 1990; Ross, 1989)
problem solving (Cobb et.al., 1991)
Value vs Quantity (Smart by Shel Silverstein) My dad gave me one dollar bill 'Cause I'm his smartest son, And I swapped it for two shiny quarters 'Cause two is more than one! And then I took the quarters And traded them to Lou For three times -- I guess he don't know That three is more than two! Just then, along came old blind Bates And just 'cause he can't see He gave me four nickles for my three dimes, And four is more than three! And I took the nickels to Hiram Coombs Down at the seed-feed store, And the fool gave me five pennies for them, And five is more than four! And then I went and showed my dad, And he got red in the cheeks And closed his eyes and shook his head-- Too proud of me to speak!
How Numbers are Used
Identifying small quantities without counting
Amounts or units determined by measuring
Numbers that name or identify
Sequence of numbers
How many in a set
Components of Numeracy by Tom Richards Students must have a strong proficiency in all of these components for mathematical literacy.
a. Forward sequences
b. Backward sequences
c. Numbers before
d. Numbers after
2. Numeral Identification
a. Single digit
b. Double digit
c. Triple digit
3. Numeral Sequencing
a. Ordering numerals
4. Subitizing a. Recognizing a set of items as a quantity 5. Cardinality a. The relationship between the numeral and its value (*** - there are 3 *'s) 6. Concept of Five and Ten (anchor numbers) a. Five and ten frames - patterns b. Combinations of five and ten
"Visualization is fundamental to learning in general and to mathematics in particular. The ability to visualize is a key component of a person's intellectual competence. It is one of the most important intellectual abilities."
Dr. Yeap Ban Har
1. Subitize - remember what you see
2. Negative Space - visualize what is missing
3. Transform an image - move it in your mind
So, what is Subitizing ?
the rapid, accurate, and confident judgments quantities of small numbers of items
http:// www.youtube.com/watch?v =hK0E1hjW2ws
Relational vs. Instrumental Understanding Instrumental: Rule: Invert and multiply. Relational: What to do and WHY ! One rectangle
Relational vs. Instrumental Understanding Instrumental: Rule: Invert and multiply. Relational: What to do and WHY ! One rectangle divided into fourths.
Relational vs. Instrumental Understanding Instrumental: Rule: Invert and multiply. Relational: What to do and WHY ! What is ? There are 2 groups of 1/4 in1/2. of the rectangle
Math Talks (Look and Talks)
Are language based
Provide varied, purposeful practice with how numbers are used
Develop part-whole thinking
Follow the C-P-C approach
Offer an avenue for communication – learning math out loud
Help students mentally count all, on, and down
"If you want your children to be intelligent, read them fairy tales. If you want them to be more intelligent, read them more fairy tales.” - Albert Einstein
Number Stories Goldilocks and the Three Bears
Number Stories The Three Little Pigs
Number bonds help students:
Understand part-whole relationships
Compose and decompose numbers
See inverse relationships
Develop algebraic thinking
Master addition and subtraction bonds (facts) through 10.
Ten is an anchor number.
Use of number bonds continues throughout the curriculum.
Build relational understanding, which is foundational for building mental math skills.
Teaching Number Bonds
Teach how numbers go together
Create addition stories
Create subtraction stories
Number bond stories
10 = + 9
Objective: Students understand and illustrate the concept of taking a number apart and then putting it back together - NOT memorize specific facts.
always start with the concrete
things that can be pulled apart, pushed back together, then pulled apart in a different way
real world bonds
students, matchbox cars, box of crayons, fruit, beans, sticks, cubes, etc.
must have similar attributes - size, shape, color
must use multiple different objects
How do you teach Number Bonds?
learn mathematical symbols
How do you teach Number Bonds?
2D representations of former concrete objects
There are __________ birds.
__________ of the birds are __________ and __________ of the birds are __________