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

2. Origins of Place Value Origins of Place Value © 2010 Herb I. Gross next Hieroglyphics to Tally Marks Lesson 1 Part 1

3. “If I have seen further than others, it is only because of my standing on the shoulders of giants of the previous generations.” next © 2010 Herb I. Gross Words to this effect were uttered by Sir Isaac Newton (1642-1727) an English natural philosopher, generally regarded as the most influential mathematician and scientist of all time. next

4. His quotation captures the essence of the development of civilization. Namely, each generation inherits a legacy from the past and in turn leaves a heritage for the future. © 2010 Herb I. Gross No place is this concept captured more elegantly than in the development of our number system. next

5. The story of whole numbers, indeed of mathematics in general, begins with what could be called the “dawn of consciousness;” i.e., the time when human beings first became aware of the thought process and consciously began to communicate with one another. To do this, language, both oral and written, had to be developed. © 2010 Herb I. Gross 1 23 456 7890

6. This in itself is a fascinating topic that we often take for granted. However, if we assume that people look for the simplest solutions to any problem that bothers them, it is not hard to imagine that the first written languages were sign languages. next © 2010 Herb I. Gross

7. next © 2010 Herb I. Gross For example… The system of hieroglyphics used by the ancient Egyptians was essentially a “sign language” in that pictorial symbols were used to stand for nouns. Often the picture was primitive (so that anyone could draw it), and it was a crude representation of the noun it represented. next

8. If, in a sign language, one wanted to indicate a horse, the person would use the symbol that represented the horse; and because everyone else knew what the symbol meant, they knew the person was talking about a horse. © 2010 Herb I. Gross next

9. If one wanted to indicate three horses, one would choose the symbol that stood for a horse and write it three times. In this context, the picture (symbol) was the noun and the number of times you drew it represented the adjective. © 2010 Herb I. Gross next

10. Using sign language presented a rather subtle problem. Because a blue pencil doesn’t look like a blue shirt, it is easy not to notice that in both cases, the adjective “blue” means the same thing. © 2010 Herb I. Gross

11. In a similar way, because three horses don’t look like three sheep, people tended to look at the adjective “three” that was modifying “horses” as being different from the adjective “three” that was modifying “sheep”. next © 2010 Herb I. Gross One might note in this regard that the English language distinguishes between large collections of animals by giving them different names.

12. For example, one makes the distinction linguistically by referring to a large number of cows as a herd, whereas a large number of sheep is a flock, or a large number of ants is an army. 1 next 1 Such a discussion links nicely with the classroom in a curriculum unit on counting. One might ask children how many other such nouns they can think of for naming collections of different animals (or, for that matter, entities other than animals). Of course, the classroom activity is not an end in itself, but rather a step on the way to achieving the desired mathematics learning outcome. More specifically, this activity is an application early in a child's education not only of the mathematical art of abstraction but the idea of tying together mathematics and language note

13. © 2010 Herb I. Gross After a while it occurred to people that if the object being discussed was “clear from context,” there was no need to draw it. next The “Invention” of Tally Marks Instead, they implicitly represented the object being discussed by a tally mark which was simply a straight line that looked like this |.

14. © 2010 Herb I. Gross Thus, to represent three horses, one would simply write | | |. And, if one wanted to represent three men, one would still write | | |. In this context “three” was still an adjective, but now it modified tally marks which in no way resembled the objects being discussed. next The point is that in going from hieroglyphics to tally marks the level of abstraction was elevated, and it had to be clear from the context of what was being discussed what noun the tally marks were representing.

15. © 2010 Herb I. Gross Keep in mind that as natural as tally marks may seem to us, it is important to appreciate that the invention of tally marks was a giant leap forward in the development of abstract thinking. next This observation takes us well beyond the confines of mathematics, and cuts to the heart of human achievement. As we mentioned earlier, each generation inherits a legacy from the previous generations, and in turn leaves a heritage for future generations.

16. © 2010 Herb I. Gross In this sense, we must remember that the contributions made by a generation must be measured as an extension of what existed prior to the advent of that generation, rather than by contributions and improvements made by subsequent generations. next

17. © 2010 Herb I. Gross next By emphasizing the human component of mathematics in student learning, as illustrated in this discussion of tally marks, a teacher can turn what might otherwise be an isolated component of the mathematics curriculum into a classroom activity that conveys to students the nature of mathematics as a chronicle of intellectual human achievement. Note

18. © 2010 Herb I. Gross While tally marks may seem (at least in comparison to our modern system of counting and doing arithmetic); primitive, they are very visual. next Tally Marks as Visual Aids Consequently they allow us to “see” properties of arithmetic which may not be as “obvious” in the more abstract notation of place value.

19. As an illustrative example, there is nothing visual about 3 + 2 and 4 + 1 that provides a clue that these two expressions represent the same number. 3 + 2 = 4 + 1 4 1 © 2010 Herb I. Gross next However, in terms of tally marks, we add numbers by “amalgamating” the tally marks that represent each number.

20. In other words, when using tally marks, one can actually visualize the addition of whole numbers, which in this example appears as… 3 + 2 = 4 + 1 © 2010 Herb I. Gross next =

21. © 2010 Herb I. Gross When translated into “plain English”, the commutative property of addition simply says that when we add two numbers, the sum doesn’t depend on the order in which the numbers are written. next next At first glance, this might seem obvious. Yet, in some other arithmetic operations, order does make a difference. For example, 3 – 2 is not the same as 2 – 3. Namely, it makes a difference whether the temperature is 3° F and then decreases by 2 ° F or whether the temperature is 2 ° F and then decreases by 3 ° F.

22. © 2010 Herb I. Gross One might be tempted to say that since 3 + 2 “looks like” 2 + 3, it is logical that 3 + 2 = 2 + 3. However, the same logic would imply that 3 – 2 = 2 – 3 because 3 – 2 “looks like” 2 – 3. next However, the use of tally marks makes it visually plain as to why 3 + 2 = 2 + 3. More specifically… Practice Problem #1 How might you use the following array of tally marks | | | | | to demonstrate why 3 + 2 = 2 + 3? next

23. © 2010 Herb I. Gross One Solution to the Practice Problem #1 This is the kind of “open ended” question, that might be answered in several ways. One such method would be to look at the following array of tally marks from two different perspectives. next

24. © 2010 Herb I. Gross One Solution for Practice Problem #1 On the one hand, we can read the array from left to right and notice that we have 3 tally marks followed by 2 more tally marks (i.e., 3 + 2). next On the other hand, we can read the same array from right to left, and notice that we have 2 tally marks followed by 3 more tally marks (i.e., 2 + 3).

25. © 2010 Herb I. Gross Is the Commutative Property Really “Self- Evident”? next Experimenting in the Classroom Divide the students into two randomly selected groups. Ask one group to find the value of 69 + 2. Ask the second group to find the value of 2 + 69.

26. © 2010 Herb I. Gross In most cases, the group that was given the problem 69 + 2 will get the correct answer more quickly than the group that was given the problem 2 + 69. next Why? We tend to add the numbers in the order in which they appear. Thus, the first group will almost immediately give 71 as the answer. They may have started by saying “69” and then counting “70, 71”. The other group starts with 2 and quickly runs out of fingers long before they count to 71!

27. © 2010 Herb I. Gross In the same way that the order in which two numbers are written can make a difference, so can the grouping of three or more numbers. For example… next By first reading the following expression from left to right and then from right to left, give two different values of the expression 2 × 3 + 4. Practice Problem #2 Answer: 10 and 14 next

28. © 2010 Herb I. Gross Solution for Practice Problem #2 next 2 × 3 + 4 6 = 10 Reading the expression from left to right, we first multiply 2 by 3 to obtain 6, and we then add 4 to obtain 10. 2 × 3 + 4 7 14 = Reading the expression from right to left, we first add 4 and 3 to obtain 7, and we then multiply by 2 to obtain 14. next

29. © 2010 Herb I. Gross next Notice that we got two different answers for the same mathematical expression. Notes on Practice Problem #2 However in mathematics, in the absence of unambiguous instructions; we have to use some sort of symbolism to tell us (without words) the order in which we are to proceed. If we had been more explicit and said for example, “Multiply 2 by 3 and then add 4”, these verbal instructions would have been unambiguous. next

30. © 2010 Herb I. Gross next To this end, mathematics uses parentheses (as well as other grouping symbols) to indicate the order in which things are to be done. Notes on Practice Problem #2 2 In this context, mathematics uses parentheses in much the same way that hyphens are used in English grammar. Notice, for example, that the phrase “the high school building” has two different but acceptable meanings. It could refer to a multiple-story building or it might be a one floor building that houses grades 9 through 12. In the former case we would write “the high school-building” and in the latter case we would write “the high-school building”. The mathematical equivalent would be to write the former as “the high (school building)” and the latter as “the (high school) building”). In particular everything contained within a set of parentheses is considered to be one number. 2 next note

31. © 2010 Herb I. Gross next Thus, to indicate that we wanted the expression 2 × 3 + 4 to be read from left to right, we would write the expression as (2 × 3) + 4. Notes on Practice Problem #2 However, when only addition is involved, the sum of three or more numbers does not depend on how the numbers are grouped. This is known as the Associative Property for Addition. If we had wanted the expression to be read from right to left we would have written 2 × (3 + 4). next

32. © 2010 Herb I. Gross next How are the two numbers (4 + 3) + 2 and 4 + (3 + 2) related? Practice Problem #3 Answer: They are equal

33. © 2010 Herb I. Gross next Solution for Practice Problem #3 Since 4 + 3 = 7 and since everything inside the parentheses is considered to be one number, we may replace (4 + 3) + 2 by 7 + 2 or 9. 3 And since 3 + 2 = 5, we may replace 4 + (3 + 2) by 4 + 5 or 9. 3 Technically, we should have written (4 + 3) = (7). However, we omit the parentheses when the meaning is clear from context. note next

34. (4 + 3) + 2 © 2010 Herb I. Gross next 4 + (3 + 2) However, by using tally marks we can make the solution much more visual and hence, easier for students to internalize. For example, with respect to Problem # 3 Notes on Practice Problem #3

35. © 2010 Herb I. Gross next Tally Marks and Geometry As useful as tally marks are for helping us to visualize some of the properties of arithmetic, it doesn’t take long for their use to become cumbersome. For example, it is difficult to distinguish at a glance the difference between eight tally marks (| | | | | | | |) and nine tally marks (| | | | | | | | |).

36. © 2010 Herb I. Gross next So it should come as no surprise that folks decided that by arranging the tally marks to form a pattern, it would be easier to distinguish one number from another. and the number nine as… In this vein, the ancient Egyptians represented the number 8 as… 3 We see this today in such places as on a pair of dice. For example, the geometric pattern of pips that name five is easily distinguished from the geometric pattern of pips that name six. In other words, while it may be easy to confuse | | | | | with | | | | | |; it is not difficult to distinguish between… and… note next 3

37. © 2010 Herb I. Gross next In fact referring to a whole number as being a “perfect square” was derived from the fact that the tally marks that represented these numbers could be arranged so as to form a square. In demonstrating this to younger students, it is often helpful to replace the tally marks by square tiles. Thus, for example, a number such as sixteen could be represented by an array of 4 rows each consisting of 4 square tiles.

38. © 2010 Herb I. Gross next That is… These geometric patterns often lent themselves to some interesting arithmetic observations.

39. © 2010 Herb I. Gross next For example, the L-shaped regions in the diagram below indicate why perfect squares are always the sum of successive odd numbers beginning with 1. Thus, for example, 16 = 1 + 3 + 5 + 7 1357

40. © 2010 Herb I. Gross next Show by using tally marks that the sum of any two odd numbers is always an even number. Practice Problem #4 Solution to Practice Problem #4 When we use tally marks to add two or more numbers we simply “amalgamate” all of the tally marks. Thus, for example, 2 + 3 would be represented by | | | | |; or since the number of tally marks doesn’t change if we shift their positions, | | | | |. next

41. © 2010 Herb I. Gross next Any odd number can be written as a group of bundles of 2 plus 1 separate tally mark. …and 13 could be represented by… Thus, for example, 7 could be represented by…

42. © 2010 Herb I. Gross next Hence, 7 + 13 would then be represented by… The remaining tally marks form another bundle of 2. That is 7 + 13 is… …and by rearranging the groups we see that 7 + 13 is…

43. © 2010 Herb I. Gross next We showed the truth of the statement only for the two odd numbers 7 and 13, but the same idea would apply to any two odd numbers. Notes on Practice Problem #4 In fact, a rather non-threatening “real life” illustration can be shown in terms of dividing candy that’s in two bags equally between two people. next

44. © 2010 Herb I. Gross next Notes on Practice Problem #4 Namely suppose we give out the candy in the usual “1 for you and 1 for you” format. When we get to the end of a bag there is 1 piece left over (that is, there were an odd number of pieces of candy in the bag). So we put that piece aside and repeat the distribution process using the second bag. Again when we get to the end, there is 1 piece left over. We may now take the two left-over pieces of candy and give 1 to each person.

45. © 2010 Herb I. Gross next Notes on Practice Problem #4 There is an unfortunate tendency to sometimes be careless with the language we use in discussing the adjective/noun theme. next For example, one can hear such things as “If you add apples and apples you get apples” or “If you add oranges and oranges you get oranges”. This could lull a student into believing that “if you add two odd numbers you get an odd number”. However, we cannot accept a mathematical statement as being true simply because “it looks as though it should be true”.

46. © 2010 Herb I. Gross next Case against Tally Marks The development of civilization may be viewed as moving from one plateau of knowledge to the next. It is as if we were traveling along a plateau where things are progressing quite nicely when suddenly we encounter a very high and steep mountain. If we are successful, we will find a way to climb this mountain and get to the next plateau. This success is a step forward, but it is the prelude to the next obstacle; namely another very high and steep mountain that isn’t even visible on the plateau below.

47. © 2010 Herb I. Gross next In this analogy, our first level was hieroglyphics, and the next plateau was tally marks. From this new plateau we were able to discover things that were not apparent at the lower plateau, but we must then be prepared to face the possibility of having to find a higher plateau. hieroglyphics tally marks new plateau

48. © 2010 Herb I. Gross next In the case of tally marks we have talked about why geometric shapes had to be introduced in order to help us recognize the difference in the amounts that are represented by two groups of tally marks. 4 In terms of our present day system of enumeration, it is clear that 1,237 exceeds 1,234 by 3. However, as we have already mentioned, we should judge the importance of an innovation by what the innovation replaces; not by what will replace the innovation subsequently. note next However, try to think of a geometric pattern that would show us at a glance the difference in size between, say, 1,234 and 1,237 4.

49. © 2010 Herb I. Gross next As a more graphic illustration of the limitation of tally marks, let’s consider the case in which the national debt of a country is $4 trillion. Because “trillion” is “just a word” and 4 is not a very “big” number we may be lulled into believing that the debt is huge but still manageable. However, this feeling might be altered if $4 trillion were translated into place value notation by writing it in the form $4,000,000,000,000. In this form, the amount looks quite a bit more ominous; even so it might still be unappreciated just how huge that debt is.

50. © 2010 Herb I. Gross next To get a better idea of the size of this debt, imagine that the nation was repaying this debt at a continuous rate of $1,000 per second. Even at this rather fast rate, it would still take 125 years before the debt was repaid! We couldn’t even begin to express what this amount was if we were restricted to having to use tally marks.

51. © 2010 Herb I. Gross next In the next part of this lesson we will discuss the next plateau that had to be reached in the journey toward how to express very large amounts. In particular, we will discuss Roman numerals and the Sand Reckoner (an early Western Civilization version of the abacus) hieroglyphics tally marks next plateau Roman Numerals Sand Reckoner