쉐도잉 연습: Addition and Subtraction of Small Numbers - 영상으로 영어 말하기 배우기
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1
Hey it’s Professor Dave; let’s talk about addition and subtraction.
2
As we have just come to understand, math never started out as a bunch of seemingly arbitrary operations meant to confuse and frustrate students.
3
It is simply that our understanding of math has become extremely sophisticated over the past few hundred years, and the fact that all of this existing and sometimes difficult math must be learned before anyone can contribute to the field is what is responsible for the way that so many people despise this subject.
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But let’s remember that math began very simply as part of our language, a set of models and symbols that helped us to communicate ideas and describe our surroundings.
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Mathematical innovations arose by necessity, and they still do, it’s simply that the frontier of today’s math lies in an abstract place that very few can understand.
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By the end of this series, maybe we can all get there, but for now, let’s start at the very beginning.
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What was the first kind of math that was developed by the human race?
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That would undoubtedly be arithmetic.
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Shortly after humans were able to count, we needed symbols to represent those counting numbers, as well as methods to manipulate those numbers in ways that represent real-life concepts.
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How many people are in the tribe?
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Two kids were just born, how many are there now?
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How many years has the tribe’s wisest elder been alive?
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Once civilizations formed and we began to trade goods with one another, we needed to be able to keep track of inventory, price items appropriately, and so forth.
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How many apples are in the basket?
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How many do you want, and how many are now left?
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Being that we like to count on our fingers, and we have ten of them, many counting systems were based on a system of ten.
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There were others based on twenty, or even sixty, but the one we use today is based on ten, so rather than diving into the anthropology of arithmetic, let’s keep our study focused on what can be readily applied.
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Presently, our conceptualization of the number ten as the basis for our numerical system is so ingrained in our collective logic that we sometimes forget that this number is completely arbitrary.
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If we had only eight fingers, things would be totally different.
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But ten is what we went with, and it works just fine.
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Getting back to the apples, the basic operations we will learn first are addition and subtraction.
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Addition is the most basic arithmetic operation, and it represents the combination of two numbers to become a single number, or a sum.
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If you get two apples from one vendor, and then three from another, how many apples did you get?
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Of course we can easily count the resulting pile and see that there are five.
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But how do we represent this mathematically?
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Using the symbols that are common of today, we would put the number two, then the addition or plus symbol, and then the number three, followed by an equals sign and then the number five.
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This is an equation, which is a statement of equality.
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The expression on the left is numerically equivalent with the expression on the right.
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This particular equation reads, “two plus three equals five”, with the word plus essentially meaning “and”, and the equals sign meaning “is”.
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Two and three is five, so five is the sum of this additive operation.
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The next operation that became necessary was subtraction, which is the inverse, or opposite of addition, in that it doesn’t find the sum of two numbers, it finds their difference.
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You bring all five of your apples home and you eat one of them.
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If one apple has been subtracted, how many are left?
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Again, it is easy to count and see that there are four left.
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But this result can also be calculated, which is much different from counting.
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We can write another equation with a five, then the minus symbol, followed by a one, then the equals sign, and the number four.
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This reads, “five minus one equals four”, which essentially means five less one is four, so four is the difference between one and five.
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On a number line, this is the distance between the two numbers, and this is an excellent way to visualize subtraction.
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Fourteen minus eleven is three, because three is the difference between the two numbers.
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It takes three to get from eleven to fourteen.
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Now that we have become familiar with the symbolic representation of these simple operations, we should discuss some applicable properties of numbers.
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Addition is commutative, in that the order in which numbers are added does not matter.
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Two plus three equals five, and three plus two also equals five.
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Subtraction is not commutative.
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It does indeed matter which number is being subtracted from the other.
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Three minus two is not the same as two minus three.
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Addition is also associative.
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This means that if performing two successive additions, the order in which they are performed does not matter.
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Two plus three plus four will be nine no matter which numbers we add first.
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We can add the first two to get five, and then add that to four, or we can add the latter two to get seven, and then add that to two.
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The result is the same.
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Subtraction is not associative.
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If we write down five minus three minus two, we could do five minus three first.
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That gives us two, and subtracting the other two, we get zero.
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If instead we do the three minus two first, we get one, and five minus one is four.
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So we can see that subtraction is not associative.
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We will learn all about the order of operations later, as well as other ways in which these kinds of properties become less obvious yet very important.
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No matter how far we choose to go with math, and no matter how complicated equations appear to be, with symbols like square roots and logarithms and integrals, we must always remember that these symbols mean something concrete.
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They are rooted in the physical world, even if in a way that is more difficult to immediately conceptualize than the plus symbol.
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One goal of this series will be to make all the other mathematical operations as intelligible and relatable as addition and subtraction.
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Once you intimately understand what mathematical constructs represent, they no longer seem arbitrary and infuriating, but instead powerful, as their utility becomes apparent.
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So let’s move forward and learn some more arithmetic, but first, let’s check comprehension.
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이 레슨의 어휘와 말하기 포인트
이 영상에는 섀도잉할 문장 62개와 단어 1107개가 있습니다. 말하는 구간의 길이는 7:37입니다. 화자는 분당 약 145단어의 일정한 속도로 말해서 섀도잉하기에 편한 속도입니다. 단어의 86%가 영어에서 가장 많이 쓰이는 3,000단어에 속합니다. 나머지는 연습 전에 미리 확인해 두세요.
이 영상의 핵심 어휘
영상에 나오는 덜 흔한 단어 15개를 발음, 뜻과 함께 정리했습니다.
- minus /ˈmaɪ.nəs/ — 빼기. Made less or reduced by (followed by an expression of number or quantity).
- symbol /ˈsɪmbəl/ (명사) — 상징, 부호. A character or glyph representing an idea, concept or object.
- mathematics /mæθ(.ə)ˈmæt.ɪks/ (명사) — 수학. An abstract representational system studying numbers, shapes, structures, quantitative change and relationships between them.
- equation /ɪˈkweɪ.ʒən/ (명사) — 방정식. The act or process of equating two or more things, or the state of those things being equal (that is, identical).
- finger /ˈfɪŋɡɚ/ (명사) — 손가락. A slender jointed extremity of the human hand, (often) exclusive of the thumb, and the corresponding part, irrespective of including a thumb, in many other…
- complicated /ˈkɑm.plɪˌkeɪ.tɪd/ (형용사) — 복잡하다. Difficult or convoluted.
- shortly /ˈʃɔɹtli/ (부사) — 금방. In a short or brief time or manner; quickly.
- innovation /ˌɪn.əˈveɪ.ʃən/ (명사) — 혁신. The act of innovating; the introduction of something new, in customs, rites, etc.
- equality /ɪˈkwɑ.lɪ.ti/ (명사) — 평등, 균등. The fact of being equal.
- inventory /ˈɪn.vən.tɹi/ (명사) — 재고. The stock of an item on hand at a particular location or business.
- pile /paɪl/ (명사) — 무더기. A mass of things heaped together; a heap.
- applicable /əˈplɪk.ə.bəl/ (형용사) — 적용할 수 있는. Suitable for application; relevant.
- dive /daɪv/ (동사) — 잠수하다. To swim under water.
- basket /ˈbæskɪt/ (명사) — 바구니, 고리짝. A lightweight woven container, generally round, open at the top, and tapering toward the bottom.
- sophisticated (형용사) — 정교한. Of a person: having obtained worldly experience, and lacking naiveté; cosmopolitan, worldly-wise.
영상에 나오는 구동사
- write down (동사) — 적다, 적어 두다. To produce or set (something) down in writing; to record something.
주의할 발음
- “th” 소리: mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, anthropology /ˌænθɹəˈpɑləd͡ʒi/, arithmetic /æɹɪθˈmɛtɪk/, logarithm /ˈlɑ.ɡə.ɹɪ.ð(ə)m/
- “sh”와 “zh” 소리: equation /ɪˈkweɪ.ʒən/, essentially /ɪˈsɛnʃəli/, shortly /ˈʃɔɹtli/, representation /ˌɹɛp.ɹə.zɛnˈteɪ.ʃən/, innovation /ˌɪn.əˈveɪ.ʃən/
- 긴 단어 — 강세 위치에 주의: mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, arbitrary /ˈɑː.bɪ.tɹə.ɹi/, essentially /ɪˈsɛnʃəli/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/
이 영상으로 연습하는 방법
- 먼저 말하지 않고 영상을 끝까지 듣고 모르는 단어를 적어 둡니다.
- 보통 속도로 한 문장씩 섀도잉하고, 화자의 리듬과 맞을 때까지 반복합니다.
- 자신의 목소리를 녹음해 원본과 비교하고, minus, symbol, mathematics 같은 단어에 특히 주의합니다.
이 영상의 문법
화자가 가장 많이 쓰는 문형을 영상 속 실제 표현과 함께 정리했습니다.
| 문형 | 영상 속 표현 |
|---|---|
| 수동태 be + 과거분사 — 누가 하는지보다 무슨 일이 일어나는지에 초점 | must be learned · was developed · were just born |
| 현재완료 have/has + 과거분사 — 과거의 일이 지금도 관련이 있을 때 | have just come · has become · has been subtracted |
| 관계절 who / which + 절 — 사람이나 사물에 대한 추가 정보 | equation, which is · subtraction, which is · calculated, which is |
쉐도잉이란? 영어 실력을 빠르게 키우는 과학적 방법
쉐도잉(Shadowing)은 원래 전문 통역사 훈련을 위해 개발된 언어 학습 기법으로, 다언어 학자인 Dr. Alexander Arguelles에 의해 대중화된 방법입니다. 핵심 원리는 간단하지만 매우 강력합니다: 원어민의 영어를 들으면서 1~2초의 짧은 지연으로 즉시 소리 내어 따라 말하는 것——마치 '그림자(shadow)'처럼 화자를 따라가는 것입니다. 문법 공부나 수동적인 청취와 달리, 쉐도잉은 뇌와 입 근육이 동시에 실시간으로 영어를 처리하고 재현하도록 훈련합니다. 연구에 따르면 이 방법은 발음 정확도, 억양, 리듬, 연음, 청취력, 말하기 유창성을 크게 향상시킵니다. IELTS 스피킹 준비와 자연스러운 영어 소통을 원하는 분들에게 특히 효과적입니다.