이번엔 교수님이 추천 해주신대로 교재의 note부분과 blog를 읽고 문제에 접근해보았다.
교재 p539 VI ‘개념 등가’ 는 어떤 진정한 개념은 잠재적인 무한 방법의 수로 정의될 수 있다는 것을 단지 의미한다. 모든 정의는 화자를 개별적인 일반성의 수준에 위치시킬 뿐만 아니라, 모든 정의는 근접한 일반성의 영역으로 어떤 이동이 있음을 암시한다.
즉, 학생에게 분수의 개념 등가를 지도하기 위해선 longitudinal, latitudinal 지도가 모두 필요하다.
blog에는 이런 내용이 있었다.
The child’s insensitivity to contradictions is just a lack of lines of longitude. Suppose you think that a number is just a numeral. Then there is no contradiction when you say that “3/2” and “1 and ½” are different numbers. After all, they ARE different numerals.
numeral 은 sign 이고 number는 meaning이다. 즉 3/2와 1and ½는 다른 sign이지만 meaning은 같음을 지도해야 한다. 위 글에서 학생은 3/2와 1and ½ 이 다른 sign임은 알고 있지만(latitude) 같은 meaning임(longitude)을 헷갈려하고 있다.
T: Look at this.(Writes "3/2" on the white board) "Three halves." Is it ONE number or TWO numbers?
S1: One.
T: Now, look at this. (Writes "1 1/2" on the white board) "One and a half". Is it ONE number or TWO numbers?
S2: Two.
T: But...aren't they the SAME number? (Writes "=" on the white board)
Ss: !!!!!
바른 개념 지도를 위해 다음과 같이 대화를 이어갈 수 있을 것이다.
T: Do you know 구구단?
S: Yes!
T: What's two times four?
S: two times..?? 무슨 뜻이지?
S: 곱하기인가
T: Yes. (Draw 2×4=)
S: Ah~ eight!
T: Good! What's four times two?
S: Four... times.. two.... 4×2 .... eight!
T: Yes. Very good. 2×4=4×2=8! 2×4 and 4×2 looks different but the answers for those two are same! Now let's try division. What's two divided by four?
S: two divided..?
T: two divided by four (Draw "2÷4=")
S: 영 쩜 오!
T: zero point five
S: two fourth
T: Yes. It's two fourth or one and a half or zero point five. It's 1/2(draw '0.5=1/2=2/4=') What else can you say?
S: 육분의 삼! three Six?
T: Good. Three sixth or three divided by six. Now, let's go back to my first question. (Draw "3/2"). Is it one number or two numbers?
S: one number
T: What about this one? (Draw 1½)
S: One number. 그러니까 둘다 같은 숫자에요!
T: Is this number(pointing at 3/2 and 1½) bigger than two?
S: No. It's between one and two.
T: Good! What about this one? (Draw 5/3)
S: One and two third?
T: And or?
S: five third.
T: Good! Now let's play the 'Don't say the word' game. Here are some cards.
T:I open one card. Aha~ Don't look! Only I can see it. Now, I will go first. three halves! next!
S1: uh? hmm... one and a half?
T: Good! Next!
S2: uh?... two and a half?
T: Wrong! Now our team gets two points. I will show you the card.
T: What's this?
S: six divided by four.
T: Yes, but! you can't say the word. You have to say three halves or one and a half... What else can you say?
S: sixty divided by fourty?
T: Very good. Now let's start the game one more time!