The Counting Numbers (Fundamentals of Math)
Ishango Bone
-22,000 years old -Found in Africa -Oldest artifact of the practice of math -tally marks
Egyptian System
-Base 10 system -order doesn't matter -staff = 1 -heelbone = 10 -scroll = 100 -lotus flower = 1000 -pointing finger = 10000 -polliwog = 100000 -astonished person = 1000000
Mayan system
-Base 20 system -Mayans used hands and feet to count -disk = 0 -place values stack, ones go on bottom
counting numbers
ordered list of numbers starting at 1
one-to-one correspondence
pairing each object with exactly one number and each number with exactly one object -last number in a set = total
number path
path labeled with successive counting numbers
subitizing
recognizing the number of objects in a set without counting them one by one
Difficulties Encountered by Children: Counting Numbers vs. Alphabet
ruins algebra for children because they automatically assume that a=1, b=2, c=3, and so on
numeral
symbol created to represent a quantity
Difficulties Encountered by Children: English
ten, ELEVEN, TWELVE, thirteen, fourteen -not many other languages do it that way, making 11 and 12 have different pronunciations
whole number
the counting numbers along with zero
unit
the distance between 0 and 1
number system we use today
Hindu-Arabic system
cardinality of a set
The number of things in a set
number line
line on which one location has been chosen as 0, and another location, to the right of 0, a=has been chosen as 1
Roman system
-Roman numerals -order DOES matter -putting a lesser number in front of a greater number means you subtract the lesser number from the number to the right of it -zero is not a placeholder ("Nulla" = none) -Romans wouldn't multiply or divide because they would instead just do a shit ton of addition or subtraction to get their answer
Base 10 System
-a place value system in which each digit is 10 times greater than the digit to the right of it -ones, tens, hundreds, thousands, ten-thousands, hundred-thousands, millions, etc
Base 10 manipulatives
-cube = ones -long = tens -flat = hundreds -block (big cube) = thousands -big long = ten thousands -big flat = hundred thousands -big cube = millions
Base 6 system
10 = 1 group of 6 and 0 ones 20 = 2 groups of 6 and 0 ones 33 = 3 groups of 6 and 3 ones -51, 52, 53, 54, 55, 100 --> 100 = 6 groups of 6 and 0 ones -These kinds of problems will have a subscript 6 when converting Place Values -1 -6 -36 -216
why is zero required in our system?
because it holds place value
scale of the number line
choice of a unit
set
collection of distinct objects
Difficulties Encountered by Children: Rote Memorization
if you make the number system a song, kids won't be able to think on their own when counting. same thing goes for the alphabet
