Q) You want to teach the children what is ‘natural’ about natural numbers, and what is ‘mixed’ about mixed numbers. Can you do it in twenty turns? Do not include learner responses EXCEPT as uptake.
T Everyone, This is Piggy.(use a puppet or a picture) He has a problem. Can you help him?
T OK. Let’s listen to him.
T (As a Piggy) I have only one orange. (Show them a real orange.) but I want to share it with my two brothers. What can I do?
S Cut!/ 잘라요~
T (As a Piggy) Oh, cut it into three parts. Thank you. But.. We want to eat it equally.
Of course, oranges don’t really cut; they pull apart into sections. And those sections are NOT usually thirds.
Three possibilities:
a) Use apples, and a knife.
b) Use abstract, diagrammatic, fake “oranges” (as pictures). Miraculously, these magical oranges have thirds, sixths and twelfths as sections!
c) Cut the oranges with a knife, for example, from the EAST pole to the WEST pole instead from the NORTH to the SOUTH poles (very messy).
You can see that each method has different advantages.
a) With the first method, we can cut TWO or FOUR apples into thirds, and it is easier to teach the difference between QUANTITY and OBJECT.
b) With the second method, we can teach sixths, twelfths, and even tenths. This makes it easier to teach about the interconvertibility of decimals and fractions.
c) With the third method, we can teach about “natural” and “fractional” numbers more graphically (and we can make an unholy mess of orange pulp and juice).
Can you cut it for us? Who can help me?
S1 (Comes to the front and cuts the orange into three pieces with hands.)
T (As a Piggy) Oh, thank you. Bye-bye!
T Bye-bye, Piggy! Everyone, how many (sic) oranges did he eat? (pointing to the one part of the orange) After S1 cut it into three parts?
Compare:
a) How much orange did he eat?
b) How many oranges did he eat?
Do they really mean the same thing?
When we teach immersion, we have to have an eye out for language teaching points as well as for the math teaching points. This is one of them! The English distinction between COUNTABLES and UNCOUNTABLES (which really lies at the bottom of the vexed problem of plurals and singulars) is really a matter of NATURAL vs. REAL numbers!
T Yes, he had one of the three parts equally divided. (writing 1/3 on the board.)
T How can you call this number in English?
T One-third. Can you repeat? One-third.
T Then how can you call this number? (write 1/4 on the board.)
T One-fourth. Good. What kind of number is this? (pointing to 1/2, 1/3)
T Fraction. (sic) '분수‘ in Korean. 'Fract' means to cut or divide. Let's repeat. Fraction. (sic)
T How about 'one, two, three..."? Are they fractions?
T NO. They are NOT cut into pieces. What kind of numbers are they?
T OK. We call them 'natural numbers', 자연수 in Korean.
T Why do we call it 'natural'? (sic)
S Nature??
T Nice try!
Remember that “Good try” and “nice try” in English are really NEGATIVE feedback. They suggest: “try again!” But they also suggest that the child is simply guessing. Is that true?
This point is also discussed in my remarks on Sunhwa’s answer. Have a look!
'Natural numbers' are not made numbers. People use it (sic) naturally, when they count apples, people, cows and so on. Repeat everyone. Natural numbers.
T Good. And what if I have three and a half apples. (write 3½ on the board.) Is it a question or a statement? Is it a fraction or a natural number?
There is some debate between immersion teachers about whether the teacher is supposed to control her language complexity or not.
There is, of course, a Krashenite wing of teachers who say that only “roughly tuned input” is necessary, so this would be perfectly acceptable, even though at sixth grade level the children could not really be required to understand the actual sentence structure of “What if I have three and a half apples, would that be a fraction or a natural number?”
But there is another wing of teachers who subscribe to the “Ouput Hypothesis” of Merrill Swain (1985). These are the teachers who are most concerned about the apparent gap between the ability of children to UNDERSTAND sentences like this and their ability to actually USE sentences like this. They point to the gap between exposure and use and ask if acquisition can really be said to be complete and whole under these circumstances.
Remember we saw that immersion teachers have to be bilingual. The reason for this is both theoretical and very practical. They have to be able to form foreign language concepts out of native language word meanings (often without using the native language, but nevertheless by knowing it) and they have to be able to answer the questions of the learners when the learners ask questions in their native language. But this principle of bilingualism can be extended to the target language too: the good immersion teacher can answer is more complex English when learners speak in simple English, and provide, if not a zone of proximal development, then at least a zone of proximal learning.
That’s why the Swain wing of teachers insisted on introducing more integrative language work in the immersion classroom, where sentences like this one (“What if I have three and a half apples, would that be a fraction or a whole number?”) can be TRANSLATED into more less complex sentences.
Now, Yukyeong has already taken the first and crucial step in doing this. She’s divided the complex, long sentence into two parts, a complex SUPPOSITION and a very simply EITHER-OR question. Vygotsky writes that every complex relation WITHIN the child (and grammar is a very complex relationship indeed) was once a set of complex relation BETWEEN people (such as complex discourse). By dividing the sentence in two like this, Yukyeong is taking a step away from complex grammar WITHIN the mind and a step towards complex discourse BETWEEN minds.
Ss A fraction?/ A natural number?
T Both are wrong. Here's a hint. A natural number and a fraction is MIXED. (sic)
“Is” mixed or “ARE” mixed?
S Mixed?/ Mix number?
T It's a mixed number, which is '대분수‘ in Korean.