Exploring Peano's Axioms in preschool,
kindergarten
(G. Peano, pronounced: Pee'-a-no,
developed a set of axioms that
describe the Natural numbers and
allows us to prove by induction)
The whole point of this exercise is to explore the
connectedness of the Natural numbers. Inductive
proofs used in mathematics or computer science will be difficult to understand
without this basic knowledge. Writing
a simple loop in a programming language uses this iteration, one after the
other, in sequence, and can be a difficult skill to grasp.
Materials:
colored stones all in one pile.
Rules:
Only one stone can be moved at a time from one pile to the next
pile.
Phase 1:
a. Show
me 3 stones
response: the
child moves one stone after the other into a new pile, verbally counting, “1,
2, 3”
Phase 2:
a. Now
show me 4 stones
response: the
child moves one more stone from the main pile into the pile of 3 stones.
b.question,
“Do I now have 5 stones? 6 stones?”
c.question,
“How many stones did I have before 4?”
“Was that 7 stones?
2 stones?”
Phase 3:
a. In
a separate pile show me 3 stones.
response: the child moves one stone
at a time into a third pile from the main pile.
b. Ok,
now we have a pile with 4 stones and one with 3.
Add the pile of
three stones to the pile of 4 stones.
response: the
child moves one stone after the other from the pile of 3 to the pile of 4.
c. How
many stones are now in a pile? 7,
so 3 + 4 is ?
Phase 4:
repeat
phase 3 but reverse the order, add the four stones to the pile of 3.
Observation: adding
3 to 4 is adding 1 to 4, 3 times in succession, and adding
4 to 3 is adding 1 to 3, 4 times in succession. The result is 7 in both cases.
Phase 5:
verbal drills, repeating aloud “3 and 4 is 7”
Remember,
these basic facts need to be known cold, NOT reasoned out,
each and every time they're needed! The
numbers themselves, can be introduced as well, “this is how we write” 1, 2,
maybe up to 9, no point introducing 0 until it's needed later.