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Phenomenological Mathematics Teaching WORKSHOP Námsstefna Flatar 29.-30.9.2006 Päivi Portaankorva-Koivisto Interactive Experiential Cooperative, collaborative Phenomenological Mathematics Teaching Exploratory Using illustrations Mathematics as a language 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) 3 5 20 2 4 : 4 6 5 3 3 3 WHY? 1 2 4 2 6 5 6 3 5 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) Exploratory Ed’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 =? 56 : 8 =? 40:10=4 3+3+3+3=12 12+2=14 14:2=7 2+4=6 50:10=5 2+2+2+2+2=10 10+6=16 16:2=8 2+5=7 The answer is 6 The answer is 7 56 sweets divided by 8 50 sweets would go well to 5 friends everyone 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 back 2+2+2+2+2=10. After that I have 16 sweets. Now I can have two more friends. That makes 7 friends altogether. Using illustrations What is the role of the artefacts? Table and Calculator? WHEN we think about Multiplication as an operation The commutativity of multiplication (Lagrange, 2006) Multiplication table and calculator 1 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!