1. The Development of Numbers The Development of Numbers next Taking the Fear out of Math © Math As A Second Language All Rights Reserved hieroglyphics tally marks Roman numerals Sand Reckoner

2. Sands of Time Sands of Time next © Math As A Second Language All Rights Reserved The Sand Reckoner The Sand Reckoner

3. We feel that students will internalize place value much better if they see it in its step-by-step development. next © Math As A Second Language All Rights Reserved For this reason, we suggest that you have the students go through the steps that preceded our discussion of the sand reckoner (hieroglyphics, tally marks (tiles), and Roman Numerals).

4. Because science and mathematics were in their relative infancy, there was no compelling need for the Romans to have invented denominations greater than a thousand. next © Math As A Second Language All Rights Reserved However, as time went on and technology increased, a need developed for expressing greater numbers. next

5. Since each new power of ten (ten thousands, hundred thousands, millions, etc.) required the use of a different letter of the alphabet, it soon became apparent that using letters could eventually become as tedious as what occurred when people had to deal solely with tally marks. © Math As A Second Language All Rights Reserved This led to the next evolutionary step in counting; namely, the invention of the abacus, or as it was known in the western World, the sand reckoner. next

6. © Math As A Second Language All Rights Reserved The sand reckoner consisted of vertical lines, arranged in a row, drawn in the sand. Each line represented a denomination that was a power of ten. onestenshundredsthousands

7. next © Math As A Second Language All Rights Reserved Stones (or pebbles) were used as “markers” or ”counters”. Thus, while a stone always stood for the adjective “one”, what denomination it modified depended on what line the stone was placed. onestenshundredsthousands

8. next © Math As A Second Language All Rights Reserved For example, if the stone was placed on the third line (from the right), it stood for one hundred. onestenshundredsthousands

9. next © Math As A Second Language All Rights Reserved 1 (a stone on the ones line) onestenshundredsthousands 10 (a stone on the tens line) 100 (a stone on the hundreds line) 1,000 (a stone on the thousands line) next

10. In the previous diagrams, we have deliberately made the stones, (dots) different colors to indicate that the color of the stone was irrelevant. All that mattered was knowing the name of the line the stone was placed on. This is quite different from what we discussed in previous presentations in which different colored disks represented different denominations © Math As A Second Language All Rights Reserved Note

11. next © Math As A Second Language All Rights Reserved To see how this is related to our adjective/noun theme, let’s analyze the diagram below… onestenshundredsthousands What we see are 3 green stones, 1 yellow stone, 2 blue stones and 4 red stones.

12. next © Math As A Second Language All Rights Reserved From the lines on which the stones appear we see that… onestenshundredsthousands Each green stone represents 1 thousand. Each yellow stone represents 1 hundred. Each blue stone represents 1 ten. And each red stone represents 1 one. next

13. © Math As A Second Language All Rights Reserved So, all in all, we have 3 thousands, onestenshundredsthousands And in the language of place value this number is 3,124. 1 hundred,2 tens,and 4 ones. 3124 next

14. The Latin word for stone is “calculus”. Thus, to “calculate” meant to do arithmetic by using stones; that is, by using the sand reckoner. 1 © Math As A Second Language All Rights Reserved Notes 1 In this context, every mathematics course that involves computation could have been called “calculus”. Thus, just as the English curriculum often lists courses by number as English 1, English 2, English 3, etc.; with the above as justification, one could have named all mathematics courses in the same way as Calculus 1, Calculus 2, Calculus 3, etc. note

15. next To include another denomination on the sand reckoner, all we have to do is add one more line. The lines we’ve drawn above take the place of our having to use I, X, C or M. 2 © Math As A Second Language All Rights Reserved Notes 2 The abacus is, in a sense, a “portable” sand reckoner. While the sand reckoner couldn’t be carried from one site to another, the abacus could. To obtain the abacus from the sand reckoner, replace the lines in the sand by wires and the stones by beads, string the beads on the wires, and enclose the entire system in a frame. As interpreted in the context of language, the wires of the abacus are the nouns, and the beads are the adjectives. note

16. next Notice that while the denominations are denoted by different letters in Roman numerals, all the lines on the sand reckoner look alike. However, the denomination represented by a line is still visible because of the position the line is in. © Math As A Second Language All Rights Reserved Notes onestenshundredsthousands MCXI

17. next Notice that as long as the denomination is visible we do not need to invent a place holder (i.e., 0). 3 © Math As A Second Language All Rights Reserved Notes 3 In fact, in our next presentation we will show why the invention of zero was the final link in the development of place value. note For example, in Roman numerals the number we would represent as 2,014 would be written as MMXIIII. There is no need to write “There are no C’s” because the absence of them tells us that none are there. next

18. The same is true when we use the sand reckoner. © Math As A Second Language All Rights Reserved Notes For example, 2,014 would be represented as… onestenshundredsthousands MMXIIII

19. next We can see from the fact that the line that represents hundreds is empty that there are no hundreds.… © Math As A Second Language All Rights Reserved Notes onestenshundredsthousands 214

20. next Addition is easy to understand using the sand reckoner. © Math As A Second Language All Rights Reserved onestenshundredsthousands Using the Sand Reckoner Stones on the same line modify the same noun (denomination). next

21. Make sure they understand that it shows 1 thousand, 1 hundred, 2 tens and no ones. © Math As A Second Language All Rights Reserved Class Activity - Sand Reckoner Have the students determine what number is represented on the sand reckoner below. onestenshundredsthousands 112 next

22. Make sure they understand that it shows 2 thousand, no hundreds, 1 ten and 2 ones. © Math As A Second Language All Rights Reserved Then, to reinforce this important fact, have them determine what number is represented by… onestenshundredsthousands 212 next

23. © Math As A Second Language All Rights Reserved Finally, using the above two diagrams, see if they can describe the addition problem that is shown below. onestenshundredsthousands 112 212 3132 + next

24. The importance of the student activity lies in the fact that adding whole numbers is a “non-issue” when the denominations are visible. © Math As A Second Language All Rights Reserved 4 And because the nouns are visible, we can write the symbols in any order that we wish. For convenience, the choice is usually to group the same denominations together and arrange them in the usual sequential order. For example, rather than write MMCXXIIIIMCCXXXII, we would usually write MMMCCCXXXXXIIIIII. note For example, to add, MMCXXIIII and MCCXXXII, we need only write the two numbers side by side; the sum is MMCXXIIIIMCCXXXII 4 next

25. The same idea is reflected in the above classroom activity. The blue dots represent the number 2,012 and the red dots represent the number 1,120. Thus, the previous diagram represents the sand reckoner version of showing that… 2,012 + 1,120 = 3,132. © Math As A Second Language All Rights Reserved In our next (and final) presentation of the development of place value, we will show how all that remains to be done is to find a way to omit having to use the lines as place holders.

26. next What happened next is the subject of our next lesson. © Math As A Second Language All Rights Reserved hieroglyphics tally marks Roman numerals next plateau Sand Reckoner