1. Mathematics as a Second Language Mathematics as a Second Language Mathematics as a Second Language Developed by Herb Gross and Richard A. Medeiros © 2010 Herb I. Gross next Arithmetic Revisited

2. Prelude to Mathematics as a Second Language next There is a “tongue-in-cheek” piece of advice to the effect that “if someone doesn’t understand your explanation of something, just explain it again; only louder”. Many students have experienced this sort of explanation too many times in their study of arithmetic; that is, listening to the same explanation being given over and over again. © 2010 Herb I. Gross

3. next To break this chain of events, we have embarked on what we believe is a fresh and productive approach. More specifically, our innovative approach to teaching basic mathematics, which we call “Mathematics as a Second Language”, is to introduce numbers in the same way that people from all walks of life use them; namely as adjectives that modify nouns. © 2010 Herb I. Gross

4. next Our technique will be to show that by using this concept, all of arithmetic can be done by knowing only the addition and multiplication tables from 0 through 9. To introduce our approach, consider the following hypothetical situation. © 2010 Herb I. Gross

5. next Suppose you are the principal of an elementary school and a mother, claiming to have a precocious 5 year old son, asks to have the boy placed in a fourth grade mathematics class. You are skeptical and decide to give the youngster a quick quiz. You say to him, “Son, how much is 3 + 2?” and the boy replies “3 what and 2 what?” Would you now discount the mother’s claim or would you place him in the fourth grade? © 2010 Herb I. Gross

6. next Our point is that the boy’s question is very important. Consider, for example, the true statement that… © 2010 Herb I. Gross 3 dimes + 2 nickels = 40 cents In this case, 3 is an adjective modifying “dimes”, 2 is an adjective modifying “nickels”, and 40 is an adjective modifying “cents”. If we omit the nouns, the above equality becomes… 3 + 2 = 40 next

7. This leads to an important result, which in this course is called the… © 2010 Herb I. Gross + Fundamental Principle for Addition + 3 + 2 = 5 only when 3, 2, and 5 are adjectives that modify the same noun. More generally, the traditional addition tables assume that the numbers being added modify the same noun.

8. next This simple principle gives us many pieces of useful information. Namely, as trivial as it may sound, we now know that… 3 apples + 2 apples = 5 apples © 2010 Herb I. Gross += 3 people + 2 people = 5 people += 3 cookies + 2 cookies = 5 cookies + = next

9. The Fundamental Principle of Addition applies even if we don’t know what the noun means. © 2010 Herb I. Gross Note For example… 3 gloogs + 2 gloogs = 5 gloogs That is, no matter what “gloog” means, 3 of “them” and 2 more of “them” are 5 of “them”! next

10. © 2010 Herb I. Gross next While this is not an algebra course, it is important to see how the adjective/noun theme can be incorporated into an algebra course in a natural and non-threatening way. For example… Practice Problem #1 How much is 3x + 2x? Answer: 5x

11. next © 2010 Herb I. Gross Solution to the Practice Problem #1 This is a more real-world interpretation of the 3 gloogs + 2 gloogs = 5 gloogs problem. Namely, in algebra, x is used as a generic name of a particular quantity. Thus, 3 of “them” plus 2 more of “them” must be 5 of “them”. That is: 3x + 2x = 5x.

12. next Obviously the numerical value of 3x + 2x depends on the numerical value of x, but the fact that 3x + 2x = 5x doesn’t. © 2010 Herb I. Gross Notes on Practice Problem #1 next The point is that too often when students are asked to find the sum of 3x and 2x, they often act as if x is just another “gloog word” and say things like, “I don’t know what x is”.

13. © 2010 Herb I. Gross Notes on Practice Problem #1 A more concrete illustration is to think of x as describing a colored poker chip. For example, we do not have to know how much a red chip is worth in order to know that the value of 3 red chips and 2 red chips is equal to the value of 5 red chips. next

14. Or on a report card, if we got five A’s, we do not usually say we got 3 A’s plus 2 A’s. © 2010 Herb I. Gross On the other hand, if we got 3 A’s and 2 B’s we do not say that we got five “AB” ’s. next Notes on Practice Problem #1 In a similar way, in algebra if A and B are names for different nouns, we cannot add the 3 and 2 in the expression 3A + 2B.

15. next As a classroom illustration, suppose a first or second grade student was confronted with a problem such as… © 2010 Herb I. Gross Because each number is represented by a ten digit numeral, this problem would be overwhelming to such a young student. next 3,000,000,000 + 2,000,000,000

16. However, suppose we recognize that in place value notation the noun “billion” is represented by nine zeroes. That is, we may translate this problem into “plain English” by rewriting 3,000,000,000 as 3 billion and 2,000,000,000 as 2 billion © 2010 Herb I. Gross next 3,000,000,000 3 billion 2,000,000,000 2 billion

17. © 2010 Herb I. Gross In this way, the relatively complicated problem 3,000,000,000 + 2,000,000,000 can be replaced by the simpler but equivalent problem 3 billion + 2 billion; the answer to which is 5 billion because 3, 2, and 5 are modifying the same noun (even though in the first or second grade the word “billion” might mean no more to a student than the word “gloog” would).

18. next The person who knows that… © 2010 Herb I. Gross Key Point 3 billion + 2 billion = 5 billion …but doesn’t know that… 3,000,000,000 + 2,000,000,000 = 5,000,000,000 …has a “language problem”, not a “math problem ”1. next 1 As a non-mathematical version of this, suppose there is a black cat in the room and you ask a person, “True or false: The cat is black?” Suppose the person answers “True”. Then you ask the same person “True or false: El gato es negro?” (Note: “El gato es negro” is Spanish for “The cat is black”). It turns out that the person doesn't understand Spanish so this time he answers “I don't know.” Now, if the person were blind, that would also explain why the answer was “I don’t know”. However, being blind doesn’t depend on language. Thus, if a person says “True” to the first question but “I don't know” to the second question the person has a language problem, not a vision problem.

19. Have you ever seen a number? next © 2010 Herb I. Gross To help you further internalize the adjective/noun theme, consider the following question… What may seem amazing to you is that no one has ever seen a number.

20. next Restated, let’s just say that we see numbers as adjectives rather than as nouns. That is, we have seen three apples, three dollars, three people, even three tally marks; but never “threeness” by itself. 3 dollars 123 123 3 apples 3 people 123 13 3 tally marks 2 © 2010 Herb I. Gross

21. next Thus, the theme of this presentation begins with the underlying principle that numbers are adjectives that modify nouns (or other adjectives) © 2010 Herb I. Gross

22. next As an application of how we can use the “numbers as adjectives” principle, consider the following situation. Many students often fail to grasp the relative size of a billion dollars with respect to a million dollars in the sense that they view both amounts as being “ very big ”. © 2010 Herb I. Gross

23. next However, suppose we now let the adjectives “million” and “billion” both modify “seconds”. Some very elementary calculations show us that a million seconds is a “little less” than 12 days while a billion seconds is a “little more” than 31 years. This gives new life to the idea that although a million seconds is quite “big”, it is still a small fractional part of a billion seconds. © 2010 Herb I. Gross

24. next It also gives students (and others as well) an easy to understand ratio; namely, a million is to a billion as 12 days is to 31 years. While it may be easy to confuse a million with a billion; no one ever confuses 12 days with 31 years! While it may be easy to confuse a million with a billion; no one ever confuses 12 days with 31 years! © 2010 Herb I. Gross

25. next However, size is relative. A quantity that is small relative to one quantity might be very large relative to another quantity. For example, while a million seconds is small compared to a billion seconds, a billion seconds is small when compared to a million days. More specifically… © 2010 Herb I. Gross

26. next Practice Problem #2 Using 400 days in a year as an estimate, how old would you have been when you had lived 1 million days? next © 2010 Herb I. Gross Answer: 2,500 years old 2500

27. next Solution to Practice Problem #2 The answer is found by dividing 1 million (days) by 400 (our estimated number of days in a year). 1,000,000 ÷ 400 = 2,500 next © 2010 Herb I. Gross In other words, if there had been 400 days in a year, you would have been 2,500 years old when you had lived for 1 million days.

28. next The fact that there are less than 400 days in a year means that our estimate is less than the correct answer. © 2010 Herb I. Gross Notes on Practice Problem #2 Thus, without having to do any further computation, we are safe in saying that you will not have lived a million days until you were more than 2,500 years old. next

29. Taking into account leap years, there are 365 ¼ days in a year. With a calculator it takes no longer to divide by 365.25 than to divide by 400. Doing this, we would find that to the nearest whole number of years… © 2010 Herb I. Gross Notes on Practice Problem #2 1,000,000 ÷ 365.25 = 2,738 next

30. The calculator is a great aid in helping us to perform cumbersome computations quickly. It is not a replacement for logic and number sense. © 2010 Herb I. Gross A Note on Using a Calculator The user has to have enough knowledge to be able to tell the calculator what to compute. For example, the calculator will not divide 1,000,000 by 365.25 unless it is told to do so! next

31. Moreover, even with a calculator we can enter data incorrectly. To guard against this, it helps to have a number sense. © 2010 Herb I. Gross A Note on Using a Calculator For example, in this discussion, our number sense told us that the millionth day since your birth cannot occur prior to the year 2500. Hence, the answer we got using a calculator (2738) is at least reasonable. next

32. The point we wish to make here is that to know the size of a quantity, we must know both the adjective and the noun it modifies. © 2010 Herb I. Gross Notes on Practice Problem #2 For example, even though a billion is more than a million, the fact remains that 1 billion seconds (approximately 31 years) is less time than 1 million days (approximately 2,700 years) next

33. In an era when education perhaps seems to be overly concerned with the use of manipulatives and high-tech visual aids, there is a tendency to forget that innovative use of language may in itself be both the greatest manipulative and the best visual aid that can ever exist; and of at least equal importance, it is available to be used by everyone. © 2010 Herb I. Gross Key Point

34. next In this course, we shall discuss how by treating mathematics as a language we can “translate” many complicated mathematical concepts into simpler but equivalent mathematical concepts. © 2010 Herb I. Gross More specifically, we shall demonstrate how by properly choosing the nouns that numbers modify, we can make many mathematical concepts easier for students to comprehend, no matter what other modes of instruction we may be using. next

35. Our previous discussion seems to demonstrate that the human mind has the ability to comprehend concepts that cannot be seen. This ability is not limited to the study of numbers. For example, with today’s technology, we can measure time to the nearest billionth of a second, yet none of us has ever seen time! What we have seen are the changes that occur with the passing of time. © 2010 Herb I. Gross A Note in Passing

36. next With our adjective/noun theme in mind, we will now begin our journey into the development of our present number system. © 2010 Herb I. Gross 3 + 2 = 5