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, Part 1 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. More specifically numbers can be viewed either as nouns or adjectives. next © 2010 Herb I. Gross

5. 0123 For example, in this case, 2 is a noun that names the point P. P next © 2010 Herb I. Gross

6. 0123 In this case, however, 2 is an adjective that modifies (measures) the distance between points Q and P. 2 PQ next © 2010 Herb I. Gross

7. However, most of us see numbers as adjectives. That is, we’ve seen… 3 people 3 apples 3 tally marks 123 123 123 next © 2010 Herb I. Gross

8. But never “threeness” by itself. next © 2010 Herb I. Gross

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

10. 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

11. 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

12. 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

13. 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

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

15. next Solution to Practice Problem #1 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.

16. 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 #1 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

17. 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 #1 1,000,000 ÷ 365.25 = 2,738 next

18. 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

19. 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 your 2,500 th birthday. Hence, the answer we got using a calculator (2,738) is at least reasonable. next

20. 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

21. 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

22. 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

23. In the next part of this Lesson, we will discuss in more detail the properties of numbers as they are used in our course. © 2010 Herb I. Gross 3 + 2 = 5