1. Development of Whole Numbers next Taking the Fear out of Math © Math As A Second Language All Rights Reserved hieroglyphics tally marks

2. Hieroglyphics next © Math As A Second Language All Rights Reserved to Tally Marks

3. The development of our number system from hieroglyphics to place value (and beyond!) is one of the greatest sagas in the development of our present civilization. next Yet, in our haste to get to the computational aspects of arithmetic, we often omit discussing this wonderful achievement. © Math As A Second Language All Rights Reserved

4. The result of this haste is that because students often are so busy learning how to perform various algorithms they fail to see the inherent beauty of how place value evolved from its very primitive origins. next What is even sadder is that if students were exposed to seeing the gradual evolution that took place in the transition from hieroglyphics to place value, they would better internalize the true power that place value brings to our ability to perform various algorithms. © Math As A Second Language All Rights Reserved

5. This clearer understanding would allow students to proceed more rapidly through basic arithmetic and soon more than make up for the time it took to discuss the evolution of our present system of enumeration. next © Math As A Second Language All Rights Reserved

6. It is beneficial for students, at as early an age as possible, to understand that people generally seek the simplest solution to any problem that confronts them, and they look for a more “complicated” solution only when the current solution becomes too cumbersome or even stops working altogether. next © Math As A Second Language All Rights Reserved

7. In this spirit, we would like students, as early as in grades K-1, to try to gain a true appreciation for the origins of mathematics by pretending to be living in the era of hieroglyphics (circa 3000 BC). next Doing this also provides a good opportunity for students to see how the adjective/noun theme is already implicitly present in the use of hieroglyphics; namely the picture is the noun and how many times the picture appears is the adjective. © Math As A Second Language All Rights Reserved

8. next To this end, have the students draw a stick figure to represent a man similar to the figure below. The students should now draw stick figures that represent three men. © Math As A Second Language All Rights Reserved

9. next Have other students draw a picture to represent an apple. Then have each student reproduce that drawing. Once the students can draw the apple, have them make a drawing that represents three apples. © Math As A Second Language All Rights Reserved

10. next Once the students are comfortable with the two drawings, have them look at a drawing in which there are 3 men and 3 apples. Lead them into a discussion in which it is clear to them that the 3 men do not look like the 3 apples, but that the concept of “threeness” is the same in both cases. © Math As A Second Language All Rights Reserved

11. next Have the students visualize the concept of “threeness” by having them use a tally mark (perhaps in the form of a popsicle stick) to replace each picture so that both the group of three men and the group of three apples become | | |. © Math As A Second Language All Rights Reserved

12. next Then discuss with them that | | | always stands for three of “something”, but we have to know from the context of what is being counted, what it is that we have three of. © Math As A Second Language All Rights Reserved

13. next © Math As A Second Language All Rights Reserved As you follow our discussion, it might seem that the use of tally marks followed closely behind the invention of hieroglyphics. However, it is interesting to note how long it sometimes takes for even a relatively simple innovation to occur after a major invention. For example, we find no mention of tally marks prior to 485 BC, some 2,500 years after the invention of hieroglyphics! Important Observation

14. next note 1 Very often a student may comprehend a concept but still be intimidated by the terminology we use to describe the concept. Thus, a youngster who knows how to share his candy by counting “2 for me, 1 for you; 2 for me, 1 for you” knows the concept of taking 2 out of every group of 3 but might still be intimidated by the phrase “two thirds” or the symbol 2 / 3. © Math As A Second Language All Rights Reserved The idea of | | | representing the number of objects in a collection is a forerunner to discussing what we mean by cardinality. In essence, a cardinal number is an answer to the question of “how many?”. So whether we are talking about 3 apples, 3 people, 3 dollars, the cardinal number is 3. (However, whether the term “cardinal” should be introduced at this early grade level is left to your own discretion.) 1 next

15. In listing the members of a set, the same member can have more than one ordinal name, depending in which direction that we are counting. At this point, at your own discretion, it might be worthwhile to introduce the students to the concept of ordinal numbers by having them demonstrate how any one of the 3 members of a collection can be listed first or second or third. © Math As A Second Language All Rights Reserved

16. next For example, in the set below, the green tile is the 4th member of the list if we are counting from left to right, but it is 3rd member of the list if we are counting from right to left. In fact, the concept of ordinal is ambiguous unless we mention how the members of a set are to be ordered. By convention, the order is assumed to be left-to-right unless stated otherwise. © Math As A Second Language All Rights Reserved

17. next Now give the students 3 tiles with different colors and have them see in how many ways the three tiles can be arranged in a row. © Math As A Second Language All Rights Reserved

18. next Make sure that the students understand that in each of the six rows, there are 3 tiles, but they are placed in different orders. For example… The blue tile is first in the top two rows of tiles, the red tile is first in the next two rows, and the yellow tile is first in the bottom two rows. © Math As A Second Language All Rights Reserved

19. In this way, they might increase their mathematical vocabulary by distinguishing between one and first; two and second; three and third; etc. next The concepts of cardinality and ordinality lead to a rule in our “Game of Arithmetic” that we have found is acceptable to youngsters, even at a very early age. © Math As A Second Language All Rights Reserved

20. More specifically, the number of members in a set (collection) 2 next In each of the 6 rows above the tiles are arranged in a different order; yet the number of tiles in each row is the same (namely, three). Of course the same result could have been demonstrated by using any number of tiles. note 2 Eventually students should be introduced to the mathematical concept of a set (namely, any collection of objects for which there is a well-defined rule for membership ). However, we feel that you should use the term that you feel best fits the student's vocabulary. next (i.e., the cardinality) doesn’t depend on the order in which we count them. © Math As A Second Language All Rights Reserved

21. Our Fundamental Property of Counting In the counting process, the number of members in any set does not depend on the order in which we list the members. next Therefore, we may state in more general terms… 123 © Math As A Second Language All Rights Reserved

22. next In terms of our adjective/noun theme, the tally marks represent what we may call a generic noun. They can represent people, apples, horses, etc., depending on what is being discussed at the time. © Math As A Second Language All Rights Reserved

23. next This leads to a subtle but important point about thinking abstractly when we use tally marks. Namely, the tally marks do not look like the nouns they modify. © Math As A Second Language All Rights Reserved Key Point Rather we have to know what the nouns are from the context in which the tally marks appear.

24. next This concludes our initial discussion of the development of our number system. © Math As A Second Language All Rights Reserved

25. next What happened next is the subject of our next lesson. hieroglyphics tally marks next plateau © Math As A Second Language All Rights Reserved