Phenomenological Mathematics TeachingWORKSHOP Námsstefna Flatar 29.-30.9.2006Päivi Portaankorva-Koivisto

Phenomenological Mathematics Teaching

Interactive Experiential Cooperative,collaborative

Mathematics as a language

Using illustrationsExploratory

Interactive

There are 4 cakes and everyone will get 3/5 of a cake.How many full servings?How much is left over?

(Tirosh, 2006)

€

4 : 35

= 4 ⋅ 53

= 203

= 6 23€

1 2

3

4

5

6

€

6 25

WHY?

Experiential

The three angles of any triangle add up to 180.

Usually like this?

But what about this? (Harel, 2006)

Cooperative or collaborative

There are two cities A and B and a railway between them. The trains are leaving the station every hour and the trip takes 3 hours.If your train is leaving at 5 pm, how many trains you see during the trip?

(Slisko, 2006)

ExploratoryEd’s Strategy (Harel, 2006)Ed is a second grader, 7 1/2 years old, and he has learned addition and subtraction. His meaning of division is as sharing equally.He was asked: ”How much is forty-two divided by seven?”

His answer was

”Forty divided by ten is four; three and three and three and three are twelve; twelve plus two is fourteen; fourteen devided by two is seven; two plus four is six.”

42 : 7 =?

40:10=43+3+3+3=1212+2=1414:2=72+4=6

The answer is 6

56 : 8 =?

50:10=52+2+2+2+2=1010+6=1616:2=82+5=7

The answer is 7

56 sweets divided by 8

50 sweets would go well to 5 friendseveryone gets 10,but then I’ll have 6 sweets left.

I gave too many each of them I should have given only 8.Now each of them gives me 2 back2+2+2+2+2=10.After that I have 16 sweets.

Now I can have two more friends.That makes 7 friends altogether.

Using illustrationsWhat is the role of the artefacts?Table and Calculator? WHEN we think aboutMultiplication as an operationThe commutativity of multiplication

(Lagrange, 2006)

Multiplication table and calculator1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100

About operation? About commutativity?

Mathematics as a language

• From where do the children learn the markings + and - before they even learn that it is mathematics?

• What do those markings mean then?

Still long way to go, but already in the move!

Thank You!