Age? Class? Subject?

[kellogg]

s1: 일반화 단계

How old is S1, Koala? Let’s start with some sense of what GRADE this is, what CLASS it is (is it English class or Math class), and so on.

Now, I didn’t tell you all this. Why not? Well, compare the following two questions:

a)    How’s the weather?

b)    How are you today?

I think you can see that a) is much less interesting than b). We can imagine that the reason is that somehow a) is an OBJECTIVE, rationalist question and b) is a SUBJECTIVE, impressionist question, and we can talk about being child-centred and invoke the names of Rousseau and Montessori and Pestalozzi and so on. We can develop a whole romantic theory around it.

But I think romanticism is unnecessary. The real answer is quite simple. With a) you will get ONE answer. With b) you will get MANY answers.

I like b). So I phrased the question in such a way that ALL of my students’ answers will be different, because ALL of them will assume somewhat different contexts: different aged children, different classes, and maybe even different language backgrounds.

But you can’t IGNORE these things, Koala! You have to GUESS. Assume. Start your answer with:

“Let’s say that these kids are ... years old.  They are in … grade in … class, so they probably know…”

 1. 사과 라는 개념을 안다

즉, 사과는 겉부분이 빨간것으로 둘러싸인, 동그란 모양의 것이고, 절반으로 자르면 안에 씨가 있는 열매이다.

절반짜리(반은 노랗고, 반은 빨간것)는 사과가 아니다.

Notice that Koala contradicts herself. If the child thinks that an apple cut in half is not an apple, then the child does NOT have the concept of an apple at all.

First of all, what Koala describes here is not a concept, but a PERCEPT. Even some person or animal who does not know what an apple is (e.g. an African jungle dweller or a penguin or a polar bear) would have the percept (e.g. would see that there is a red, round shaped object cut in half with a seeds inside).

Secondly, the teacher says “How many APPLES?” and the child replies. What the child could be responding to is “How many…?” and not to the specific word “apples”. The child could assume that the question refers to the plate of objects.

Thirdly, the word “apple” in Korean is not really the same concept as “apple” in English. “Apple” is the “fruit of fruits” for English speakers, and even the “object of objects” (e.g. the thing that fell on Newton’s head). Persimmons are probably the “fruit of fruit” for Korean speakers. In English we talk about crabapples and horseapples and apple pie as the essence of American culture. There is no such equivalent in Korean.  

But it doesn’t really MATTER, Koala. Because the REAL question here has nothing to do with apples at all—the issue is ONLY whether the child understands what a mixed number is, and how it is related to whole numbers, on the one hand, and fractions, on the other (i.e. how it is both linked to and distinct from both kinds of numbers).

So the problem is NOT whether the child knows what an apple is. The problem is whether the child knows what a MIXED NUMBER is and how it is related to objects on the one hand and known concepts (e.g. fractions and whole numbers) on the other. That’s the problem.

2. 사과라는 개념을 만족하는 '사과의 갯수'는 1개이다.

Remember that you have to show THREE things in your answer.

a)    You understand that there’s a difference between school learning (what the child says in response to a teacher question) and mental development. So for example, the child says things in respond to a question that the child might not normally think. Would the child NORMALLY think that an apple cut in half is not an apple?

b)    You understand that the zone of proximal development (the “improvement” that the teacher wants to bring about in the child’s understanding of mixed numbers) is about DEVELOPMENT and not just about school learning. So for example NEITHER child has the ability to use mixed numbers in this example, and the teacher has to somehow split the difference. How does the teacher do that?

c)    There is a difference between what people say in a second language and what they actually think. So for example the child might know perfectly well that there is not ONE apple or TWO apples on the plate but something in between but the child is not able to say it in English.

'Number' is a quality of a group of objects we call "quantity'라는 개념을 사용할 수 있다.

Yes, I think I wrote that. But you must forgive me, Koala; I get carried away sometimes. I just wanted to say that at a given stage of thinking the child’s idea of number is equivalent to COUNTING (that is, to an activity you do with your fingers, or with words) and at another stage of thinking the child’s idea of number is a kind of invisible TRAIT of a group. I’m afraid the way I expressed it was a little Hegelian. Think of it like this:

a)    Number = pointing

b)    Number = counting

c)    Number = grouping

d)    Number = quantity

e)    Number = relator, operator, relation, operation

왜냐면 , one whole apple을 가르켜 사과라고 했으므로

Good! Koala says that the child’s idea of number is that ONLY whole numbers are numbers. Now, maybe this is true, and maybe it’s not true (see below, the problem of equating what children say in a foreign language with what they actually think).

But it’s a good guess, and it’s a good place to start. The child has a concept of number, and it’s roughly the concept of number that we find among prehistoric people or just among ordinary people in the Middle Ages, that is, people who have not been to school: the naïve notion that numbers are just properties of groupings, or quantities of objects. Therefore “zero” is not a number, and neither is a fraction. Negative numbers are not numbers either.

Now, what about S1’s friend, S2? Let’s find out.

3. 하지만 추상화(abstraction) 단계는 아니다.

Better! Koala says that what is missing is ABSTRACTION. That suggests a good way of answering the LAST part of the question, that is, what should the teacher do (or what should the teacher ASK) to help the child to form the concept of mixed numbers?

The teacher has to get the child to ABSTRACT the quantity from the actual objects.

사과라는 개념을 제거하고 순수하게 1+1/2)이라는 pure idea를 가져와야 되는데,

1/2사과는 사과가 아니라고 생각하기 때문이다.

Koala INSISTS that the child thinks that half an apple is not an apple. OK. What about S2? Does S1 simply think that S2 is WRONG?

(구체적인 사물(object)을 제거하고, 순수한 추상적 아이디어만 빼낼 수 있는 능력은 없다)

=>즉 1/2이 숫자(number)를 의미한다는 것을 모르며,

1사과가 아닌것은 사과로 취급하지 않는다.

Koala says (I think) that the child is having trouble removing the abstract idea of quantity from the object. That I think is COMPLETELY correct, and in fact I think it’s the absolute KEY to the whole problem.

In Korea, and in the USA, we teach the concept of number through COUNTING OBJECTS. This makes it very easy to introduce number very early on. But it ALSO makes it very DIFFICULT to teach fractions and part-whole relations, and almost impossible to teach algebra.

Now, in Russia, and also in China, we teach the concept of number through MEASURING. This makes it rather difficult at first—the child has to think about his age, for example—How old are you? Are you EXACTLY six years old? Who is older, S1 or S2? How many months? In other words, how many twelfths older? What is your age in DECIMALS? How about HEIGHT? And WEIGHT?

Now, you can see that BOTH methods use the child’s own experience as a starting point. If anything, the Soviet method is actually MORE child-centred, because it actually uses the child’s own body and mind (although of course children do use their fingers when they count). But there is a BIG difference in the result.

In New York, Jean Schmittau (2004) the Soviet method to teach first graders about quantity. Like this:

a)     The children learn to compare quantities of objects and say which is bigger.

b)    The children are required to compare objects which and say which is bigger

c)     The children are required to compare objects which cannot be moved.

d)    The children measure each object and compare quantities of measurement units.

This makes it easy for the children to learn “part-whole” relationships as the basis of equations. For example:

                                         15

                                         ∧

                                       3  12

Learning equations like this makes it much easier to introduce algebraic statements:

                                         15

                                         ∧

                                       3   ?

s2: naming 단계

눈 앞에 보이는 구체물을 가리키며,  하나,둘,셋 이런식으로 세는 단계이다.

'사과란 무엇인가'에 대한 정의를 내리지 못하기 때문에

눈에 보이는 대로, 닥치는 대로 갯수를 센다.

Now, I don’t actually see any pointing or naming in the data, Koala. So if you want to ASSUME this, you are going to HAVE to begin by saying something like:

“Let’s say that these kids are ... years old.  They are in … grade in … class, so they probably know…”

If you do this, though, you are going to have to EXPLAIN why the two children give different answers. Does it mean they are at different levels of development? Or does it mean that they are at different moments of learning? What’s the difference?

Ah—that brings us to the REALLY HARD part of the question. Here we go!

a) you understand the distinction between " development" and "learning"

발달은 생각이 사회화된 결과다.

사회화의 응집된 형태 중 하나다. 따라서 비자연 발생적 개념이다. 곧 어린이 생각에 이질적이라는 것을 의미한다.

발달은 사회화된 생각이 어린이 자신의 생각방식을 몰아낼때만 발생가능하다.

어린이의 자연발생적 개념은 극복되어야 한다.

반면, 교수학습은 일반화, 추상적 체계를 갖춘것이다.

구체적 현상적 체계를 갖추지못한 것 사이의 상호관계가 전개된다.

다시말해 교수학습은 이 둘 사이의 전환을 촉진한다.

즉, 교수학습은 발달과 상충된 개념이다.

하지만 전적으로 상충되는 것은 아니며 "일정한 기준"(->사회화를 통해 비발생적 개념을 어린이에게 심어준다.

그러면 비발생적 개념은 원래 어린이가 갖고 있던 스키마 및 배경지식과 결합되어 아이의 개념형성 체계를 확장해준다)

에 비추어본다면, 서로 상호작용하며 발달한다.

=>교수학습을 통해 비개체발생적 개념발달이 일어난다. 

Koala argues that the child’s thinking needs to be SOCIALIZED, and that generalization and even abstraction are not inherent qualities of child thinking. They have to be brought in from the outside, through constraint, compulsion, constriction.

So teaching the child about numbers is largely a matter of SOCIALIZATION. This is actually VERY consistent with her analysis of the data, where she does NOT see any elements of conceptualization or abstraction in the child’s thinking and in fact sees NOTHING but water, without a single dot of wine. The child does not even have the concept of “apple” much less mixed number.

But take a look at this:

2-5-8] 이제 우리가 관심을 가지고 있는 기간 동안에 펼쳐진 발달 경로를 간결하게 기술하고자 한다. 도식적으로 말하자면, 우리의 가설은 아래와 같은 방식으로 우리가 발달의 전체 경로를 제시하도록 요구하고 있다고 말할 수 있다. 말의 최초 기능은 의사소통기능이고 사회적 연결 기능이고 그리고 성인이 그러한 것처럼 아동도 그럴 수 있는, 자신의 환경 속에 있는 사람에게 작용하는 기능이다. 그래서 아동의 최초의 말은 순전히 사회적인 것이며, 사회화된(социализированной) 이란 용어는 어떤 것이 시초부터 사회적이 아니라는 그리고 그것의 변화와 발달 과정을 통해서 그렇게 되었다는 관념과 연결되어 있기 때문에, 아동의 최초의 말을 사회화된 말이라 칭하는 것은 부적절한 명명이 될 것이다.

2-5-9] 이후에만, 성장과정에서, 다기능적인 아동의 사회적 말은, 개별적인 기능으로 분화한다는 원리에 맞게 발달하고 특정 연령에 이르러서야 충분히 파악할 수 방식으로, 자기중심적 말과 의사소통적 말로 분화한다. 우리가 위에서 이미 지적한 이유뿐만 아니라 앞으로 보게 될 이유 때문에, 즉 말의 이 두 형태가, 우리 가설의 관점에서, 모두 다 말의 사회적 기능이지만 다른 방향을 향한 말의 기능으로 보인다는 것 때문에, 우리는 피아제처럼 “사회화된(socialisée)”이란 표현으로 말의 그 형태를 명명하기보다는 의사소통적 말이라 명명하기를 선호한다. 따라서 이 가설에 따르면, 자기중심적 말은 사회적인 것에 근거하여 출현하고, 이 때 아동은 행동의 사회적 형태를, 즉 집단적 협력의 형태를, 개인의 심리적 기능들의 영역으로 전이시킨다.

2-5-10] 아동이 이전에 사회적 형태이었던 행동과 똑같은 형태의 행동을 자신에게 적용하려는 이런 경향은 피아제에게 너무도 잘 알려져 있고, 논박으로부터 아동에게 반성(размышление)이 출현하는 것을 설명하기 위해 그에 의해 자신의 책에 적절하게 사용된다. 아동의 반성은, 반성의 발달을 낳는 기능적 계기들을 드러내는, 다투는 과정에서 출현하는 토론에서, 이 단어에 가장 적합한 뜻인, 아동의 집단적 논쟁 후에 나타난다는 것을, 피아제는 충분히 설득력 있는 방식으로 보여주었다.

It looks to me like Koala has given us a completely PIAGETIAN answer. Of course, that’s perfectly OK—I understand that Piaget is what we teach and learn at 교대.

But what about Vygotsky? Is he just wrong, Koala? Why do you think so?

There’s another problem, Koala. If you insist on the Piagetian view, which is that the child can only acquire academic concepts through FORCE, through CONSTRAINT, and through COMPULSION (e.g. “socialization”), then it’s very hard to use the concept of the zone of proximal development. And that’s the next hard part!

c)    zone of proximal development-

Take a look at the LAST part of 6-4, Koala. Or the explanatory note on p. 489.

Vygotsky begins with the well known problem of determining “MENTAL AGE” and notes that the well-known solution is the Binet-Simon tests (fore-runner of the IQ tests of today, and cousin of the sort of tests that Thorndike was working on). He says that these tests, since the child does them alone, can only gives us the matured mental functions of the subjects and cannot tell us anything about their maturing mental functions. In order to discover the developing mental functions, he suggests tests where the children are given Binet-Simon problems to solve and then offered leading questions, partial solutions, or demonstrations. The difference between their non-autonomous and their autonomous performances can be said to reflect the next zone of development, because what the child does with assistance today will be “intra-voluted” and become an autonomous capacity tomorrow (see Chapter Four).

Vygotsky goes on to link this difference between non-autonomous and autonomous performance not only with the child’s ontogenetic development but even with phylogenetic development; he mentions Kohler’s finding that apes, for example, can only imitate within the very specific limits of ape intelligence, and that we can find those limits simply by looking at the error curve. Where the given task is within the ape’s zone of proximal development, we find the error curve decreasing almost immediately, and where it is not we find that the ape can be taught to perform complex operations but only with many errors. Vygotsky notes that not one of the three theories linking teaching-and-learning to maturation and development (the Piagetian, the Thorndikean, and the Koffka-esque) has been able to offer a clear criterion for both linking together and distinguishing between human education and animal training, but that the zone of proximal development offers just such a criterion. The ape, like the human, can only INTELLIGENTLY imitate what he can understand; the ape, unlike the human, cannot understand human culture.

 Vygotsky now draws CONCLUSIONS from all four research series.

1 It is already widely felt, among teachers that the teaching-and-learning of writing, for example, should not simply wait for the development of the undergirding skills and mental functions; writing is, after all, an important source of those skills and functions itself. This correct feeling now receives THEORETICAL support: “Pedagogy must orient itself not to the yesterday of development but to its tomorrow.” He turns his back on instruction through complexes, and criticizes (by name) the pedologists, including of course himself and calls for leaving “complexes” outside the school room door, like Mary’s Little Lamb.

2 HOWEVER…Vygotsky ends this section with a warning. While it is true that teaching a child only what he already knows how to do is a waste of teacher’s and learner’s time, it is equally true that teaching a child materials which correspond to a zone of development which is NOT the next one is also a waste of teacher’s and learner’s time. A child who cannot read or write cannot be taught alongside children who already can, because literacy is not a “competency” but a whole new stage, or rather line, of development. What the zone of proximal development really describes is not learning, but the very mode of learning. The mode of learning itself develops, and is quite specific to every stage of the child’s maturation. Because at school age this stage has been clearly linked to conscious awareness and mastery of the system of concepts, the problem academic concepts and their system is the central problem of teaching-and-learning and development for school children.

c)어린이가 외국어로 말하는 것은 사실 그들이 생각하는 것과 완전히 같지 않다.

Yes, exactly! So it seems to me…we CAN’T really say that the child has no concept of “apple”, or that the child thinks that an apple which is cut in half is not an apple.

BUT we can say that the child doesn’t know how to express mixed number concepts in a foreign language. How can we teach the child?

We know that the word is almost always ready when the concept is ready. Now…is the concept ready or not?