The Abstract Concrete

[kellogg]

3. The Abstract Concrete (The L1 curriculum)

a) Getting Attention: How does a teacher get children to create concepts for which there are no words?

b) Giving Information: Do children directly learn new concepts or do they reinvent or reverse engineer them? How?

c) Checking Integration: Should we teach counting or measuring as the basis of quantity?

d) Let’s Look Back: Some answers

e) Let’s Look Forward: Some new questions

a) Getting Attention: How does a teacher get children to create concepts for which there are no words?

Ms. Shin Jisu is not an immersion teacher. She’s a regular teacher, and she works with a foreign language teaching assistant. They take turns teaching the regular English curriculum. The language teaching assistant is good at “getting attention”, so Mr. Price usually teaches “Look and Listen”.

But as Kowal and Swain pointed out, this kind of “look and listen” very often leads to SEMANTIC processes: the children understand a few words and simply guess the rest. This leaves them unprepared for SYNTACTIC processing, where you not only have to know the main words but also the functors (articles, prepositions, and so on) and you have to know their word order.

So today Ms. Shin is following up with a more production oriented activity. This will, she hopes, give the children an opportunity to use the main words (the content words) that they have just learned. But it will also PUSH them to use functors and create sentences that they have never heard before.

It is March. The lesson topic is “I like spring”. But Ms. Shin is looking forward to the MAY lesson, which is “When is your birthday?” So she is personalizing by getting the children to say whether their birthdays are in spring or summer; whether they were born fall flowers, or winter blossoms. Then she writes their names on a calendar pie chart, which looks like this:

T: (after asking and answering about the students' birthday months and putting their names accordingly on the sheet on the board) OK, so, have a look. So, uh,.,with Mr. Price, you guys are learning 4 seasons, right? So, we have from January to December, but what season do you think this is? (pointing to around the spring picture)

Ss: Spring!

T: Spring.

This is nothing but “Initiate-Respond-Eval‎uate”, which is the archetypical structure for classroom exchanges outlined by Sinclair and Coulthard (1976) and Mehan (1979). The teacher initiates with a question to which she already knows the answer to that she can eval‎uate the response. But this time the TEACHER is bored, or perhaps Ms. Shin is just feeling a little mischievous. She teases the children, like this:

T: Are you sure?

Ss: Yes.

T: (writing 'spring' at the right-top corner) Yes. It is spring. And after spring, what comes?

Ss: Summer!

T: Summer comes. (writing 'summer' at the right-bottom corner) And then,,,?

Ss: Fall./ Autumn.

T: Fall (writing) or...?

Ss: Autumn.

T: Normally British people tend to say Autumn instead of Fall. But American people usually say Fall. And it is, finally?

Ss: Winter.

Lyster (1990: 166) complains that learning language in a classroom situation is a narrowly “functional” form of bilingualism that often gives a wrong impression of how language is used outside the classroom. In many situations outside the classroom (say, at a bus stop, or in a restaurant) people ask questions when they want to GET information. But in classroom situations, including immersion classrooms, teachers ask questions when they want to GIVE information.

Here, however, no actual new information has been given in the whole of this long exchange. Instead, the teacher really does want to GET information. But it is information about how much information the foreign teacher was able to give the children; information about where the lesson should actually start.

Where? Here! Because it is March, it isn’t really spring yet. But it’s not exactly winter either. Outside the snow is long gone, but the spring buds have not yet appeared, and there is a harsh, dry, cold wind blowing in from Central Asia so they are not likely to appear for a while. In Korean, there is a very good expression‎ for this: “꽃샘추이”. But what do we say in English? “Indian winter?”. No, that’s not it. 

T: Winter. (writing) But listen! I don't like winter because it is TOO cold. (pointing to the winter at the left-top part) And I also don't like spring because yellow dust is in the air. (pointing to the spring) So,, I like, I like,,,, over here (underlying 'wint' in the winter and 'ring' in the spring) I like WINTRING! (wrinting 'wintring' between the winter and the spring)

In (at least) three different ways, Ms. Shin is taking the high steep road.

a)       Ms. Shin is using artificial concepts. Of course, they are not COMPLETELY artificial; they are made of “natural concepts”. But they are not the natural words that English speakers use.

b)       Ms. Shin does NOT try to map English words onto a Korean set of meanings as the children did with 온도기 (“thermometer”). She does not call her artificial spring-summer period as “The-spring-wind-envies-the-tender-buds-of-spring” or anything like that.

c)       Ms. Shin puts the words into her portmanteau word in CHRONOLOGICAL ORDER. Although “sprinter” would be a little more “English-sounding” than “wint-ring” she uses “wint-ring”, she doesn’t use it. She wants the children to go FORWARD, not back. because her real goal is to get kids to make up her own words.

T: Umm! I don't like spring because of yellow dust, I don't like summer because there are TOO many mosquitoes. SO! I like,,,?

Ss: Sumer./ Sprimm./ Sprimming./ Sprmm./ Sprumm./ Sprummer./ %#$^@@ (and more inaudible words)

T: What?

S1: Sprummer.

Ss: (a couple of inaudible words)

T: Sp,Sp,Sprummer? OK. I like Sprummer? OK. It's up to you. (writing 'Sprummer' between the spring and the summer) What about here? (pointing to between the summer and the fall) Summer is TOO hot and in Fall, I feel lonely because leaves fall, so,,,,any good ideas?

Ss: Supery/Summal./Summery./ Autterfall./ Supering/ Auffal/ Summal $#@%! (all are not so clear)

T: Su,,,Su,,,(underlying 'sum' in the summer and 'all' in the fall.) not SummER and SummAL? Summal? Yeah, possibly. (writing 'Summal' between the summer and the fall) What about....here? (pointing to between the fall and the winter) I feel lonely and it's too cold.

Notice that the teacher BEGAN by using the idea that winter was too cold and spring was uncomfortable because of the yellow sand to search for a perfect compromise, “Wintring”. She then said that the period between summer and fall was better than either. But now the teacher points to the place between fall and winter and suggests that it’s too lonely AND too cold. But the children have figured out what to do, and they ignore this apparent inconsistency.

Why? Because the teacher was simply using her likes and dislikes the same way that she used the familiar words of “spring”, “summer”, “fall” and “winter”, as a way of BOOTSTRAPPING the new concepts. This technique, which we can name after the way in which a computer uses one system (the operating system) to start up and another one (the application) to do actual work, is similar to the “planned spontaneous” discussion noted by Lapkin and Swain (1996: 246) in the work of skilled immersion teachers in Toronto. The discussion is planned insofar as the teacher intends to use it to bootstrap the main task but it is spontaneous insofar as the teacher does not know how long this will take and it can go on anywhere from a few minutes to half an hour.   Once the kids have grasped the SYSTEM behind the new concepts, the “likes” and “dislikes” motivation is no longer necessary.

As you can see, bootstrapping is a useful technique, particularly in language teaching where we often have to use ONE kind of meaning to get at ANOTHER. For example, when we point we use gesture to bootstrap object-related meanings, and when we name and define we use grammar to get at more abstract meanings). Bootstrapping in this way is like the electric starter in a car, or the kindling in a fire; it is a lower level means of meaning making that is used to initiate a higher level one. The children have now seized the concept, and they produce cognate concepts very readily. Look:

Ss: (they get louder)Wintal!/ Faunter!/ Founter. / Faunter!/ Falter./ Father!/ %$@#^@ (a lot of inaudible words)

T: But it goes from Fall to Winter, so maybe from here...

Ss: (keep creating inaudible words) Founter./ Falter.

T: Falter? OK, (writing 'falter' between the fall and the winter) So, unm, Jinho's birthday is in,,,between Spring and,,Sprummer. What about your birthday, when is your birthday? Well, Obama's birthday is... (we had done a guessing game last lesson and it was about guessing some celebrities' birthdays.)

S2: August.

T: In....

Ss: Summal.

T: Summal? Summal? (laugh) What about you? Can you tell me about your birthday?  When is your birthday? (S3 raises her hand)

S3: My birthday is in Wintring.

T: Ahh,,what month? When, when is,

S3: March.

T: Your birthday is in March so maybe your birthday is in Wintring. Or,,, what else? 지원?

S4: My birthday is in Falter which is December.

T: Ahh, December? So Falter? It could be Falter? What about you back there? Yes!

S5: 어,,My birthday is in Winter, uh, January.

T: Ah, it is January? So it is totally Winter.

You can see that one effect of this kind of interpolation of new seasons is to make the old seasons more GRADEABLE; they are less like nouns and more like adjectives.

Here’s what Ms. Shin says about the lesson as a whole:

‘In this data, we can see that the teacher and the students are approaching two different kinds of concepts: everyday concepts and artificial concepts. Everyday concepts refer to concepts that are naturally formed in everyday life. In this data, for example, the words such as spring, summer, fall and winter belong to everyday concepts. On the other hand, artificial concepts are formed through fabricated conceptual process.

‘Portmanteau words such as brunch, which is the combination word of breakfast and lunch, are good examples of artificial concepts. Here the teacher and the students are blending the sounds from two season words to imply "in-between" seasons. (T: "I like wintring!' / S1: 'Sprummer.') According to Vygotsky, everyday concepts are contrasted with academic concepts which refer to concepts that are created from formal and abstract verbal definitions through curricular planning in school instruction (Vygotsky, 1978; 1987; 1994; 2009).

‘Artificial concepts resemble academic concepts in that both of them stand in contrast to everyday concepts; while artificial and academic concepts are volitionally created from abstract verbal definitions, everyday concepts are acquired from natural and concrete use of them spontaneously during practical play-like activities. (Shin Jisu, 2009)’

Ms. Shin is right. The two kinds of concepts, the everyday concept and the more systematic academic one, look the same, and many people, e.g. Goodman and Goodman (1990), and Lave and Wenger (1998) have treated them as identical. But there are dangers in treating them as the same.

First of all, we can see from just this short example that bootstrapping concepts can be very teacher centred, and it is time and energy consuming for that reason; teachers have to build a kind of “bridge of questions” between the types of concepts and it is hard to keep coming up with initiates. Secondly, bootstrapping concepts can easily become a substitute for the main activity rather than a means to get there quickly: teachers can spend the whole hour reviewing the previous lesson or talking about likes and dislikes instead of actually teaching the lesson. And thirdly, there are curricular dangers: our current English curriculum, for example, is very heavily oriented towards “Look and Listen” and “Listen and Repeat”, and as a result we don’t get very much syntactic processing done in class.

This problem occurs across the curriculum. For example, in math class we often teach the idea of QUANTITY by teaching counting. This means that we will teach NATURAL numbers (1,2,3…) before we teach REAL numbers (1, 3/2, 2, 3, 3.141…). The idea is that the natural numbers will bootstrap the children’s understanding of quantity. But in some ways this way of beginning makes it rather difficult to teach fractions because the children think that fractions are not really quantities, and of course it makes it very difficult to understand irrational numbers and limits.

One of the reasons why Ms. Shin successfully uses bootstrapping in this data without falling into any of the traps is that she has a clear idea of where she wants the children to go and what she wants them to do. On the other hand, she is also open to the actual concepts that they are able to form. This allows her to integrate the new concepts with the old, in such a way that they are clearly linked as well as distinct.

Of course, the teacher has to keep BOTH the practical activities that make up the lesson and the teaching concept in mind! This isn’t actually as hard as it seems; in fact, it’s NOT as hard as keeping, say, the vocabulary teaching points in mind when you are teaching grammar, or keeping “Look and Listen” in mind when you are teaching “Let’s Play”. In fact, this is one place where immersion makes things EASIER for the teacher rather than more difficult.

For example, in math teaching we often begin, as the Americans do, with COUNTING.

One banana, two bananas, three bananas, four!

Five bananas, six bananas, seven seven bananas, more!

The concept here is, of course, NUMBER. But we find the same concept in one of the most difficult teaching points we have to teach in English, namely the definite article “a”, “the”, and “some”.

The usual way we conceptualize this (and the way M.A.K. Halliday conceptualizes it) is as a kind of menu of choices.

Articles

Indefinite

Definite

“The”

Plural: “Some”

Singular

Before a vowel: “an”

Before a consonant: “a”

Now this tells us the structure of the language as a SYSTEM, and if we were computer programmers following a flow chart it might be very useful. But there are some things that make it less useful for teaching.

First of all, it’s very complex, and it covers too much. It’s quite independent of the SUBJECT MATTER that we are talking about: science, math, ethics, and science for living: Secondly, because it’s not attached to specific concepts it has to be remembered separately from the actual concepts, and this makes it hard to use. Thirdly, it doesn’t help the children with one of the main problems they have, which is to understand why “a banana” turns into “the banana” once you mention it.

a) T: Do you like bananas? (concept)

b) T: Here’s a banana! (example)

c) T: Whose banana? My banana or your banana? Who wins the banana? (object related meaning).

Here we can see that the choice of article is not simply part of an abstract system, and that “a” and “the” are not just equivalent choices. They are FUNCTIONALLY different. They have different FUNCTIONS, the way a boat is used for one thing and a car for another. “A” is a number, like one. “The” is a demonstrative, like “this” or “that” or “these” or “those” or even “there” and “then” (time demonstratives).

Because our children are getting to the point where they can distinguish an idea from a thing, we can even show them that in English we use the plural for the idea, as if we were talking about a BIG group of bananas:

a)       T (holds up one finger) : Look! Here’s a banana! (example)

b)       T (holds up two fingers): Look! Here’s a pair of bananas!

c)       T (holds up three fingers): Look! Here are some bananas!

d)       T (holds up five fingers): How many bananas?

e)       T: A LOT of bananas! I like bananas!

Notice that the word “bananas” in b), c), d) and e) is exactly the same. But only e) is really a general concept (that is, the IDEA of bananas). The other words are referring to actual objects. It’s almost as if English speakers are cave men: they have a number “a” for one, a number “pair of” for two, and anything more than two is “some”. If we want the idea of the banana, we just use “a whole lot of bananas”. Korean, of course, is very different.

In this chapter we will try to take Ms. Shin’s approach far beyond her lesson and even far beyond English to the development of concepts generally. We will argue that when we integrate old and new concepts in this way, we don’t just “build the new concepts on top of the old ones”, because the two types of concepts are actually growing in opposite directions: one of them from concrete to abstract, and the other from abstract to concrete. For that reason, it often helps to start out a lesson, or even a whole unit, with the concept in mind rather than just the particular activities that the teacher has to teach.

b) Giving Information: Can children directly take over new concepts ready-made? Or do they reinvent or re-engineer them? How?

First, we’ll look at some work by Paula Towsey and Carol MacDonald in South Africa. Then we’ll take a closer look at the actual writings by Vygotsky that it’s based on. Finally, we’ll return to the math classroom for a moment and look at a model immersion lesson by Ms. Kim Eunshil which is based on using MEASURING instead of COUNTING.

Carol McDonald was, like Merrill Swain, one of the first eval‎uators of a programme of immersion in South Africa which had to work under the old apartheid system, where blacks and whites went to separate schools and even lived apart in separate “countries” (one very rich, large, and continguous and other unspeakably poor, tiny, and broken up into “independent homelands” that were sometimes not much bigger than a large township).

English offered a great deal to the struggle for a new South Africa. First of all, it offered black children an alternative to Afrikaans, the language of the apartheid government. Secondly it helped to unify the many different peoples of the “independent homelands” into a nationwide movement for “People’s English”. Thirdly, it offered them a voice in international affairs.

English also offered a lot to the children. First of all, it offered them academic concepts, a wide range of new words in geography, science and history that were not really available in their own languages. Secondly, it offered a new self-consciousness: it allowed them to consider their own experience independently of their own languages, and to access parts of it in a conscious and analytical manner. For example, the stress on MAP MAKING allowed the children to integrate their local knowledge with the science concepts being offered in immersion. Thirdly, it offered metalinguistic consciousness. Just as a map makes it possible to look at your home village from a whole new angle and seeing it as part of a much larger country, a foreign language offered them a way of reflecting on your mother tongue, the realization that this world of words which appeared to be coterminous with your social reality is really nothing more than one single instance of the human ability to make languages.

The South African experience has a great deal to offer US as well. First of all, it’s a FOREIGN language situation (not a second language), it’s ENGLISH (not French), and it’s mostly ELEMENTARY school level (not middle or high school). That’s us. Secondly, it’s a milieu where immersion has to be informally implemented rather than as part of a government policy; where it is tolerated rather than encouraged. That’s us too! And finally, there is an explicit connection between immersion and the developmental level of the children, specifically, the development of conceptual thinking, which is often taken for granted in other immersion studies.

There is even a rather mysterious connection between one of the founders of South African immersion, Len Lanham, and Vygotsky himself. Lanham traveled to the USSR in the early 1960s and learned the methods of Professor B.V. Belyayev. Belyayev, a professor of language teaching methodology at Moscow University, apparently worked alongside Vygotsky from the inception of the Vygotsky school (and completed his Ph.D. work on an ESSENTIAL Vygotskyan theme, the unity of emotion and cognition in language learning).

Like Vygotsky, Belyayev believed that speech “bootstraps” thinking in development (rather than the other way around, as appears to happen when the process of development is complete). Words, and in particular word meanings, are what give rise, eventually, to concepts. Belyayev  notes that this raises the interesting question of whether one really thinks the same way when one speaks another language, and if no, is one really the same person?

Belyayev distinguishes between various different types of word meanings, some abstract and conceptual (and thus differently mediated by different languages) and others concrete and object related, and thus identical as long as the objects are present (1963: 146). He concludes that at the very least one may think in very different concepts (1963: 49, 64-65), at least some of the time.

It would appear that immersion teaching, taking place in a relatively context-impoverished classroom which simply doesn’t have all the objects we need to talk about, needs to build on the former, and not the latter. But this really ignores the key problem of how children bootstrap conceptual knowledge from perceptual knowledge.

We often see problems like this in real data in English class. For example:

T : OK. Let's play. Everybody, stand up

T : Now, everybody, do as I do. 자, 선생님 하는 대로 따라하세요. Everybody, look and copy me.                       

Ss : Copy? copy, copy...

T : 선생님 하는 대로 하세요.

T : It's time to do the dishes. (pretending to do the dishes) 

Ss : It's time to do the dishes. ( copying the teacher)

T : Everybody, move, move. 

The children do not appear to understand the phrase “do as I do”, and it doesn’t make it any easier to collapse the complex grammar into the word “copy”. So it seems at least unlikely that the actions of the teacher should mean the concept “It’s time to do the dishes” to the children. More likely they simply experience this as a concrete activity, that is, something like “Listen and do” (or even “Look and do”).

So how can children create abstract concepts from the concrete percepts around them? For example, if I hold up a picture like this:

And I say: “Look! This is lag!”

How do the children know that I mean the pig, or the hang-glider, or the act of hang-gliding (or even the picture or the card)?

That’s the question that Towsey and MacDonald tackle. In order to tackle it, they turn to an experiment done by one of Vygotsky’s students, Sakharov. Sakharov created a set of blocks that looked like this:

(cev)

(mur)

(bik)

(lag)

You can see that the first two rows of blocks are small and the last two rows are relatively large. You can also see that the first and third rows are short, while the second and fourth rows are thick or tall (this is rather poorly indicated in the picture, but it’s clear with actual blocks). This gives us four artificial concepts something like Ms. Shin’s artificial seasons:

But you CAN’T see this very well when the blocks are all mixed up, particularly when the blocks are all different colors and so many different shapes. There is, however, a secret “word” written on the bottom of each block: cev, mur, bik, and lag. The child tries to sort the blocks into four categories and by turning them over one by one figure out what the secret word really means.

Here is the secret!

QUALITIES	+ DIAMETER	- DIAMETER
+ HEIGHT	Lag	Mur
- HEIGHT	Bik	Cev

After the children have figured out what the four words mean, the children have to apply these new words to a set of glasses and candles (some of which are tall and fat, some of which are tall and thin, some of which are short and fat, and some of which are short and thin) like the children in Ms. Shin’s class.

Now, Hanfmann and Kasanin, who were the first to use this test outside the USSR, were interested in diagnosing schizophrenia. Towsey and MacDonald are more interested in what Vygotsky and Sakharov were interested in: not the difference between schizophrenics and normal people but rather the difference between children and adults, and especially the difference between elementary school children and adolescents.

In other words, like Vygotsky before them, Towsey and MacDonald are interested in ONTOGENESIS, in CHILD DEVELOPMENT, and not simply microgenetic learning. That is, they want to see qualitative changes in the nature of the word meanings and not simply the addition of new word meanings. In order to see this, they take a very big cross section of subjects, ten three-year-olds, ten five-year-olds, ten eight-year-olds, ten eleven-year-olds, ten fifteen-year-olds, and then adults, including a 76 year old man.

Here’s the result of the transference test:

You can see there is a really BIG leap (nearly sixfold!) between the five year olds and the eight year olds, and a slight decline between the fifteen year olds and the adults. But of course, that is the transference exercise only. If we use Hanfmann and Kasanin’s system and award points for fast solutions and penalize wrong guesses, here is what we get:

Group one is the three-year-olds, group two is the five-year olds, group three the 8 year olds, and groups four, five, and six are 11 year olds, 15 year olds, and adults respectively.

The problem with looking at the results in this purely quantitative way (as a kind of IQ test) is that it makes concept formation look very smooth and linear, almost like the child’s growth in height or weight. For example, it looks like the overall Hanfmann-Kasanin score DOUBLES between the three year olds and the five year olds. On the other hand, the increase between the eight year olds and the eleven year olds looks like it is only on the order of a third, and the increase between eleven year olds and the fifteen year olds (that is, the “transitional age”, when children really start thinking in concepts, at least according to Vygotsky) looks like only about 50%.

The transference scores are a little less misleading! The three year olds have ZERO transference and even among the five year olds we get a group score of less than 15% of the total score. A few short years later, though, that score is over 50% and by fifteen the score is actually marginally higher than that of the adult group.

When we look at the actual process of problem solving and not simply the result, we can easily see why the overall Hanfmann-Kasanin score is so misleading. Sure, there are big differences. But the differences are very different. The three year olds simply ignore the words and even the sorting task, and play with the blocks, making towers and houses and so on:

But the five year olds have the opposite problem; it’s hard to keep them from peeking at the words and cheating, so anxious are they to get the sorting right! With older children there is a very strong tendency to seize on some very salient property when sorting (color and shape are the obvious ones). For example, this subject tried to site according to color, even though there are only FOUR groups.

It is only when the children look at the bottoms of the blocks and check what words are there that they find that the necessary concepts are not available from direct perception or even from their own language; they must SYNTHESIZE the idea of height and diameter to arrive at a correct solution.

Now, this does NOT happen all of a sudden. Instead, there are a large number of intermediate stages in the child’s reasoning, and this is why the experiment is interesting to us. According to Vygotsky, the experiment can tell us a good deal about the way children use word meanings before they learn conceptual thinking.

Like many Soviet scientists Vygotsky believed that experiments involved the empirical working out of a theory. The experimental procedure itself is closely related to the categories found in Hegel’s logic:

Section One: Being

Quality, that is, color and shape. In this case, green, orange, white, yellow and triangle, circle, half circle, hexagon, trapezoid.

Quantity, that is, number and group. In this case, four groups of five or four blocks

Measure, that is, height and diameter. In this case, tall or short, big or small.

Section Two: Essence

Ground, that is, the contrast between a figure and a background. In this case, the contrast between the blocks and the board, and the contrast between one block and other blocks.

Appearance, that is, the contrast between various features in a single block. In this case, the contrast between color and shape, or shape and height, or height and diameter.

Actuality, that is, the contrast between each unique block and all of the others.

Vygotsky’s interpretation of the data corresponds roughly to the third section of Hegel’s logic:

Section Three: Concept

Subject, that is, the “I”, the ego, the being, in this case the child,

Object, that is, the “it”, the alter, the thing, in this case, the block

Idea, that is, the idea that an “it” is a kind of “I”, the idea of a being in the thing, in this case, it’s the idea that there is a human idea, an idea that exists first for others and then for me, in the measurable qualities of various quantities of block (size and diameter)

Sure enough, Vygotsky divides his data into three “steps” which we have translated as 단계 or stages, and subdivides them into “substeps” which are unfortunately also sometimes called “steps”.

Stage One: Heaps, or Jumbles (purely subjective groupings)

The random heap (blocks are simply put together by the subject for no objective reason, the stage of ‘this’ and ‘that’)

The spatial heap (blocks are put together by the subject because they happen to be together, the stage of ‘here’ and ‘there’)

The two-step heap (blocks are put together by the subject into random or spatial heaps, and then a block is selected from each heap and put into a new jumble)

Stage Two: Complexes (objective, that is, concrete, perceptual, factual groupings)

The associative complex (blocks are put in a group which is based on association with a prototypical “model”, e.g. a triangle with a triangle, or red with red, or tall with tall)

The complex-collection (blocks are put together in groups were every member is different in some way, e.g. triangle with trapezoid with circle with semicircle, or blue with red with black with green)

The chain complex (one block leads to another, e.g. a blue triangle leads to a red triangle leads to a red square leads to a green square)

The diffuse complex (one trait is selected, e.g. color or shape, however this trait is allowed to vary in an unbounded way, e.g. yellow to green to blue, or triangle to trapezoid to square)

The pseudoconcept, or bounded complex (the trait selected is defined in a way that seems externally similar to an abstract concept, e.g. color or shape, but is in fact based on factual resemblance rather than an abstract understanding of the quality, e.g. “this looks kind of like that”)

All of the preceding can be considered “pre-concepts”, which is how Vygotsky appears to refer to them in Chapter Six. By helping the child to GENERALIZE and to ABSTRACT traits from concrete objects, they set the scene for.

Stage Three: potential concepts and concepts

Everyday concepts (these are used syntagmatically and not as part of a paradigmatic hierarchy of generality, e.g a triangle is considered unrelated to a circle)

Scientific concepts (these are used in school and as part of a paradigmatic hierarchy of generality, e.g. a circle can be considered, like a triangle, to be a type of polygon with infinitely many sides that are infinitely small).

Vygotsky is careful to point out that higher concepts are not necessarily better: it is just as wrong to use science concepts for everyday purposes as it would be to teach everyday concepts for science purposes. Sometimes we want to refer to circles as dinner plates and full moons (say, when we are teaching a lesson about 추석) and sometimes it is better to speak of it as a polygon with infinitely many sides (when we are teaching geometry or calculus, for example).

Because the two kinds of concepts refer to the same reality, they are linked as well as distinct. In fact, Vygotsky believes that they eventually grow into each other: we learn that the twenty-four hour day (with the international date line) and the twelve-hour (“am” and “pm”) clock are one and the same thing, and both are useful when you make international telephone calls.

Because everyday concepts and academic concepts form what Vygotsky calls a “complex unity”, they grow in opposite directions: the one from concrete to abstract, and the other from abstract to concrete. And this is why Davydov developed a theory of the primary curriculum based on the idea of “rising to concrete”. Here, the curriculum is not organized around particular activities (such as counting) or even particular themes (such as geometrical shapes) but instead around solving problems which require the use of a “germ”, a “kernel idea”, a basic concept (e.g. “height + diameter”).

But of course in Vygotsky and Sakharov, and in Towsey and MacDonald, the child is in a “strange situation” (Bronfenbrenner). The child is not really alone in solving problems (because the child has the researcher, and above all the tools that the researcher has prepared), but the child is not in a classroom, and the child is working with what Ms. Shin Jisu called “artificial concepts” rather than socioculturally generated ones.

Vygotsky’s experiment did not try to replicate the way children form concepts in classrooms, working with a teacher. He was merely trying to show that young children do not simply take on adult word meanings directly (“like a flock of roasted pigeons which drop out of the sky into the child’s open mouth”, as he says). Nor do they merely “reverse engineer” them from finished adult concepts.  But they do not just rediscover them entirely on the basis of their own thinking and their own errors (as Piaget believed) either.

The reason why concept formation in the L1 curriculum is so complex is that all three of these processes are intertwined (and sometimes tangled) with another process: the child’s gradual movement from a purely syncretic (that is, subjective) approach based entirely on the child’s actions to a naively realist (that is, objective) approach based on concrete, perceptible actions to a recognition of the objectiveness of abstract ideas.

But this is based on Vygotsky’s experimental work, not his empirical work. So it doesn’t necessarily answer our question about how children develop concepts in the L1 classroom, much less how they do it in L2. In later, empirical, work (which we will look at in the next chapter) Vygotsky took the child out of the cage of this “strange situation” and studied the school child in the wild, that is, in his/her natural habitat—the classroom! But to end this chapter, we too need to “rise to the concrete”: we need to look at a specific problem in math teaching, namely whether math teachers should begin teaching the concept of number with COUNTING or with MEASURING.

c) Checking Integration: Should we teach counting or measuring as the basis of quantity?

Here is an immersion “model lesson” based on Lesson One of the sixth grade math book, “Decimals and Fractions”.

T: Good afternoon, everyone. How are you today?

Ss: I'm very good. How about you?

Notice that the children say “I’m very good”.

a)       Why do they say “I’m” and not “we’re”? Does this suggest an orientation towards CLASSROOM discourse or NON-CLASSROOM discourse?

b)       Why do they say “I’m good” rather than “I’m well” or “I’m fine”? Americans say this (perhaps because they are generalizing from “How’s life?” or “How are things?”). But to say “I am good” might, without context, be interpreted as an ethical or metaphysical statement. Does this suggest an orientation towards CLASSROOM discourse or NON-CLASSROOM discourse?

T: Good.

What is good? The form of the answer or its content?

T: Today we're going to study lesson 1.(pointing to the number 1 on the board)

T: Let's read the title of lesson 1 altogether. Ready, go. (Pointing to the letters next to the Korean sign '단원‘)

Ss: Decimals and fractions.

T: Everybody, look at the screen. This morning Seungyun and Gangseo had a fight. (pointing to the TV screen) Everybody, let's watch TV and let's find out what the problem is.

-Dialogue-

Seungyun: Hello, Gangseo.

Gangseo: Hi, Seungyun.

Seungyun: How long is your ribbon?

Gangseo: My ribbon is 2/5 meters. (showing a number card) How about yours?

Seungyun: My ribbon is 0.3 meters.

Gangseo: Umm. My ribbon is longer than yours.

Seungyun: No, my ribbon is longer than yours. Look at..0.3m.

Gangseo: 2/5m. I'm longer than yours.(sic) No. No.

T: Why, why do you think they fight each other? (sic) Did you find it? (sic)

What does the teacher mean to say? Why can’t the children answer? Look to see WHO answer’s the teacher’s question.

Ss: ...

T: Seungyun and Gangseo insisted their ribbon is longer than other's (sic) How long, how long was Seungyun's ribbon?(Opening her arms to show 'length') Do you remember?

The teacher answers HER OWN question (incorrectly). Notice that she immediately concentrates on QUANTITY rather than COMPARISON (“How long” rather than “Who is right?”) What would some of the advantages be of doing it the OTHER way (“Who is right?” rather than “How long”)?

Ss: 0.3 m.

T: Yes. Seungyun's ribbon was 0.3m.(putting the number card on the board) What about Gang seo's? Gangseo's ribbon, how long was it?(Opening arms to show 'length')

Ss: 2/5m.

T: 2/5m.(putting the number card on the board)

T: (pointing to the “2/5” card and asking) This is the decimals or fractions? (sic) (pointing to the title on the board)

Suppose we wanted to ORGANIZE this along the lines of DIRECT-INFORM-CHECK. Can you use a COMMAND to direct attention, a STATEMENT to provide information (“two/fifths” or “two divided by five”) and a QUESTION to check integration?

Ss: Decimals.

T: Is it decimals? (sic)

Ss: Fractions.

T: Okay. 2/5 is Fractions and 0.3 is ...(pointing to the letter 'decimals')

Ss: Decimals

Why does the teacher use the PLURAL here?

T: They cannot compare them. So, who can help them? Whose one is longer? Do you know the answer?

S1: 2/5.

T: 2/5 is longer than 0.3. How? How do you know that? How do you know that? Okay. You don't know the reason but you know the answer?

S2: To make fraction, oh! make decimals! (sic)

T: You mean 2/5, change 2/5 to decimals (sic)?

S2: (Nodding)

T: Good idea. What else?

S3: Make 0.3 to fractions (sic).

T: Very good.

Notice that our teacher is having BIG problems with the plural. When we want to express a general idea, we DO use the plural. That is why the TITLE of the lesson is plural. But when we rise to the concrete, we need a singular example! Can you find the places here the teacher simply overgeneralizes from the title of the lesson and change them? Like this:

T: (pointing to the “2/5” card and asking) This is the decimals or fractions? (sic) (pointing to the title on the board)

à

T: (pointing to the “2/5” card and asking) Is this a decimal or a fraction?

This is a SIXTH grade lesson. If we go back ONE year, we find that the children are learning the notion of quantity by counting oranges. Oranges, of course, are very countable. But unlike apples they are also FRACTIONABLE, that is, they can be taken apart into segments without a knife. How could we use the oranges to introduce the idea of fractions?

In Russia, Davydov (1990), who was a disciple of Vygotsky, created a method of mathematics teaching which began with the idea of measurement rather than counting. He saw, correctly, that beginning with measurement makes it easier to teach children that fractions and decimals (and even infinite series of decimals) are also numbers; in other words, beginning with measurement means that the children learn REAL numbers instead of the so-called “natural” ones (which are not particularly natural!)

In New York, Jean Schmittau (2004) used Davydov’s 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   ?

Let’s imagine you to show the children that fractions and decimals are connected because they have a common underlying concept, namely part-whole relations. You do this:

T: Let’s see. Gangseo’s ribbon is long. How long? More than one meter or less?

T: Good. Now, one meter is FIVE fifths. Gangseo’s ribbon is how many fifths? Two fifths or three fifths?

T: Right. So…ONE meter minus TWO fifths is how many fifths?

                                       Gangseo                                        Seungyun

                                       1 meter (5/5)                                  1 meter (?)

                                         ∧                                                ∧

                                 2   ?                                               3  ?

What does the teacher say about Seunyun’s ribbon?

Now let’s imagine that our teacher wishes to teach multiplication, in order to multiply Gangseo’s fifths by two and get Seungyun’s tenths. Instead of teaching multiplication as repeated adding, Davydov treats it as a PROCESS, an OPERATION.

e)       The children measure a quantity of banana milk in a bowl using a cup. (e.g. three cups)

f)         The children begin to measure a very LARGE quantity of banana milk in a tank using a very SMALL cup. (e.g. 24 cupts)

g)       One child, tired of the task, suggests using the bowl instead of the cup.

h)       The children then calculate like this:

                                              CUP

                                      ↙                ↖

                                 3 cups                     3 x 8 = 24

                                 ↙                                     ↖                      

                    BOWL  →           8 bowls  →          TANK

Notice that the activity is GOAL ORIENTED, like a good game. When we begin games with some sense of the overall GOAL (e.g. “Let’s play ‘Find the Banana’” rather than “Let’s play Bananajumanji”). In the same way we can begin the child’s concept formation by aiming the concept towards a concrete goal (e.g. “Let’s measure the banana milk!”). We can call this goal oriented phase of the activity the SUBJECTIVE phase. Notice that language we use here will often begin with “Let’s…” or “I want…” or “We need…”).

Notice also that the activity requires concrete OBJECTS. When we begin games we often do so with an introduction of the PARTS (e.g. “Look! This is a card! This is a spinner! This is a board! This is a prize! What’s this? Yes, it’s a banana!”). In the same way we can begin this activity with concrete objects (e.g. “Look! This is banana milk! A whole TANK of banana milk! And here is a CUP.”). We can call this thing oriented phase of the activity the OBJECTIVE phase. Notice that the language we use here will often begin with “This is a…” but it will continue with “the …”.

Finally, notice that the activity is a complex process made up of two sub-processes, each of which requires an OUTCOME. When we teach games we often do so with the introduction of MOVES (e.g. “Now, I put THIS card HERE. I spin the spinner! I throw the dice on the board. I move my horse ONE, TWO, …. How many squares?”). In the same way, we can show the children how actions with objects produce particular RESULTS (e.g. “Now, I fill the cup with banana milk. ONE cup, TWO cups… How many cups?”)

Now, let’s imagine that the teacher wants to use this method to explain HOW Gangseo and Seungyun can change fractions to decimals.

T: Let’s find out! Gangseo has TWO fifths. How many tenths?

What can the teacher say to change Seungyun’s tenths to fifths?

e) Let’s Look Back

Remember, we asked these questions.

a) Getting Attention: How does a teacher get children to create concepts for which there are no words?

b) Giving Information: Do children directly learn new concepts or do they reinvent or reverse engineer them? How?

c) Checking Integration: Should we teach counting or measuring as the basis of quantity?

We saw that Ms. Shin got her kids to create concepts for which there were no words by providing only ONE example. Because the new concepts were related to old concepts, and because they were also related to a SYSTEM (the calendar) the children were able to integrate the new concepts into the old concepts but also to GENERALIZE and reorganize the whole system using the new concepts on the basis of a single example. This ability to generalize to a whole system is, according to Vygotsky, generally true of academic concepts, and it’s one big difference between academic concepts and everyday life (we cannot reorganize the calendar in this way and expect to be understood outside the classroom). It’s a clear strength that we have in immersion teaching, but not in regular English teaching.

We also saw, from Towsey and MacDonald’s reverse engineering of Vygotsky and Sakharov, that children do not learn new concepts directly, nor do they reinvent or reverse engineer them. Instead, concept formation is a complex process in the child’s long term development, involving a subjective phase of syncretic thinking, an objective phase of complexive thinking and only then an abstract concrete phase of conceptual thinking.

Finally, we applied these three stages to a concrete problem, namely the problem of teaching quantity on the basis of measuring which arises in the first lesson of the sixth grade math book. (We noted that using the Davydov method, this problem arises in FIRST grade!) In this way, the teacher can REVERSE ENGINEER ontogenesis: we begin with a subjective phase, by announcing the GOAL, we proceed with an objective phase, by manipulating OBJECTS, and what emerges is the necessity of a new CONCEPT.

f) Let’s Look Forward

But what happens when all this happens in a foreign language? Are the concepts which are built up in this way identical to the concepts in the native language or are they slightly different? We already saw that the way in which the concept of a noun (e.g. “bananas”) is expressed in English differs significantly from the way it is expressed in Korean; we use the plural and then we use articles to give examples. Verbs are even trickier.

What happens when we put Korean concepts into English? Will a child who learns concepts in two languages really think like two different children, as Belyayev suggests? Or will the child be able to see that foreign language concepts and native language concepts, although they LOOK a little different, are really two instantiations of the same concept? These are some of the problems we’ll be looking at in the next chapter, which you remember is devoted to the support we provide in immersion for the native language.

References:

Belyayev, B.V. (1963). The Psychology of Teaching Foreign Languages. Oxford, London & New York: Pergamon.

Davydov, V.V. (1990) Types of Generalization in Instruction: Logical and Psychological Problems in the Structuring of School Curricula. Reston, VA: National Council of Teachers of Mathematics.

Fleer, M. (2006). ‘Meaning-making science’: Exploring the sociocultural dimensions of early childhood teacher education. In K. Applet‎on, (Ed.) Elementary Science Teacher Education. Mahwah, NJ: Lawrence Erlbaum. 107-124.

Goodman, Y.M. and Goodman, K.S. (1990) Vygotsky in a whole language perspective. In L.C. Moll (Ed.) Vygotsky and Education: Instructional implications and applications of socio-historical psychology, Cambridge: Cambridge University Press, 223-250.

Lapkin, S. and M. Swain (1996) Vocabulary teaching in a grade 8 French immersion classroom: A descriptive case study. The Canadian Modern Language Review/La Revue canadienne des langues vivantes, 53 (1) 242-256.

Lave, J. and E. Wenger (1991) Situated learning: Legitimate peripheral participation, Cambridge: Cambridge University Press.

Lyster, R. (1990). The role of analytic language teaching in French immersion programs. The Canadian Modern Language Review/La Revue canadienne des langues vivantes, 47 (1) 159-176.

Mehan, H. (1979) Learning Lessons: Social organization in the classroom. Cambridge, MA and London: Harvard University Press.

Nuttall, C. and D. Langhan (1990). The Molteno Project: A case study of immersion for Enlgish medium instruction in South Africa. In R.K. Johnson and M. Swain (Eds.) Immersion Education: International Perspectives. Cambridge: Cambridge University Press. 210-238.

Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy. European Journal of the Psychology of Education. 19 (1) 19-43.

Sinclair, J. McH. and Coulthard, R.M. (1975) Towards an Analysis of Discourse. London: Oxford University Press.

Shin, J.-S. (2009) Unpublished thesis work.

Towsey, P.M. and MacDonald, C.A. (2009). Wolves in sheep’s clothing and other Vygotskian constructs. Mind, Culture, and Activity. 16: 3 234-262.