이번엔 교수님이 추천 해주신대로 교재의 note부분과 blog를 읽고 문제에 접근해보았다.
Good! I’m glad you find the stuff useful, rather than just confusing.
We saw earlier that difficulty is often associated with CHOICE. Existence questions are easy because they involve relatively few choices (“yes/no”) while essence questions involve more (“what?”) and explanatory questions involve far more choices (“why?”).
I am never really sure how much CHOICE to give my students. I like a LOT of choice, but I know that in a foreign language it can be very intimidating! I guess the thing to do is to wait until there are specific problems to solve, and then list the choices.
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.
“The EQUIVALENCE OF CONCEPTS simply means that any true concept can be defined in a potentially infinite number of ways: just as the number 1,000,000 can be expressed in an infinite number of ways mathematically, we may define an apple as a fruit, a commodity, a plant product, a living thing, a phenomenon of chemistry, of physics, etc.) Every definition not only situates the speaker at a particular level of generality (e.g. “fruit”, “commodity”, “plant”) but suggests certain moves to proximal areas of generality (e.g. “plant”, “item”, “tree”).”
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.
Right! The relationship between a numeral and a number is actually very similar to the relationship between a word and a meaning, between pronunciation on the one hand and communicative function on the other.
Good! You can see that the arithmetic operations are really LATITUDINAL moves. But “=” allows us to move LONGITUDINALLY, by showing that, for example, a particular whole number belongs to a larger and more general set of non-whole numbers.
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
Notice that Hongkong takes SERIOUSLY my advice about considering the difference between foreign language and mother tongue.
The children are using a special “plane” of mother tongue speech to talk about the plane of foreign language speech. This is an externalization of some of what happens internally. But how much?
This gets to be particularly important on Question Three of the final. Which plane (‘outer speech’, ‘inner speech’, ‘thinking’, ‘feeling’) involves a transition from Korean to English?
Koala asked this question once. This is our chance to ANSWER!
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!
Good—notice the use of S-T translation. This is not discussed very much in discussions of TEE. But it seems to me a crucial fact of classroom life. Just as T-Everyone questions produce a kind of model of HOW to answer a question (by showing which answers live and which must die), we can see that S-T translation creates a kind of model of a “mother tongue” plane and a “foreign language plane”.
T: zero point five
S: two fourth (plural or singular?)
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? (plural or singular?)
T: Good. Three sixth (plural or singular?) 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.
Notice that BOTH the children AND the teacher have some trouble with plurals. It should be “five THIRDS”.
Where does this problem come from? Is it a SPEECH problem or a THINKING problem? Is it really a problem, or does it mean a HIGHER, OPERATIONAL way of thinking about fractions (that is, thinking of a fraction as a relationship between two numbers rather than as a specific quantity)?
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!
Notice that there is NO problem with plurals when we use everyday concepts such as “halves”. Can you explain why?
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 forty?
T: Very good. Now let's start the game one more time!