Teaching Math Across Cultures: Connections to the Maya
Maya Math WebQuest
Decoding the Dresden Codex:
Which Way Do We Go?
For this part of the WebQuest, you will need to see a copy of a selected part of the Dresdent Codex. There is an image of it at the bottom of this page and extra paper copies available in the materials box in the classroom if you want a copy to write on.
Now you can begin to figure out what a whole section of numbers in the Mayan codex might mean. Look at how the numbers appear on the pages of the codex.
You probably notice that most of the groups of dots and bars appear in clear rows and columns.
Archaeologists always choose the clearest areas of information to begin their work, so let’s begin with the large group of digits (dots and bars) just below the center of the first page of the codex.
Right away, you face another problem. You can read the individual digits. You even know that they have place values. But how do you know which digits go together to make a number? There are six columns and three rows. Which way do you read? Left to right? Right to left? Top to bottom? Bottom to top? The best way to answer these questions is to choose your best hypotheses and test them. It often helps to hear other people’s thoughts when you reach this point, so if you’re working with friends, discuss your ideas with them.
If your group is large enough, different people can try different
experiments. For instance, one group can look at the symbols as three six-digit numbers read from left to right. Another group can read the same digits from right to left. A third group can read the symbols as six three-digit numbers read from top to bottom. A fourth group can try reading the six three-digit numbers from bottom to top.
Remember, you are working in base twenty, so the first digit is in the 1s place, the second digit is in the 20s place, the third digit is in the 400s place, and so on. Record all the different results, then see if one set of numbers makes more sense than the others.
If you read the symbols horizontally (either right to left or left to right), you came up with some huge numbers. After all, the sixth digit in base 20 is the 3,200,000s place! So that’s probably not the correct interpretation.
If you read the digits up and down, you came up with more workable numbers. Check to see if you got the following results.
Reading from top 1s to bottom 400s (and left to right):
Reading from bottom 1s to top 400s (and left to right):
Take some time to study these numbers. Can you discover any order or pattern in either of these sets of numbers?
You might have noticed that the numbers read from bottom to top change in a more regular pattern. They get larger from left to right, and they change at a fairly steady rate. Do you think a set of numbers in a pattern is more likely to carry useful information than a more random set of numbers?
What Does It All Mean?
Now you’ve figured out that the Maya counted in base twenty. You’ve also discovered that they wrote and read their numbers from the bottom up, and from left to right. But what were they writing about in this codex? How would you begin to answer that question?
Look at the list of numbers you just deciphered, reading from bottom to top. One way to explore a series of numbers is to find the difference between each pair of numbers:
6,151 – 5,934 = 217
6,328 – 6,151 = 177
6,545 – 6,328 = 217
and so on.
Check to be sure you got these differences:
217 177 217 177 188
What could they mean? The archaeologists who studied the Dresden Codex found an important clue when they added any two of the numbers together. See what happens when you do that.
All of your answers should be between 354 and 434. One of the sums is 365. Does that remind you of anything?
Each of the numbers you added is close to half of 365—the number of days in a year.
An astronomer would also recognize the number 177 as exactly six lunar months of 29 1/2 days each.
It turns out that this part of the Dresden Codex is a record of astronomical observations made by the ancient Maya. This text gives the timing of eclipses of the moon, which occur about every half year.
One More Problem:
You may be wondering why all five numbers do not divide exactly into lunar months. To begin to solve the problem, look at the bottom of the same page on the codex. Another easily identifiable group of numbers is there. Use what you have learned to discover what those numbers are.
Check to see if you got these numbers:
177 177 177 177 177 148
Two of these numbers match the differences above, but three do not. This is because Maya mathematicians used two different numbering systems. For their everyday accounting needs, they used the standard base twenty with its succession of powers: 1, 20, 400, 8,000, and so on, as we have done so far. When working with astronomical calculations, however, they used slightly different bases: 1 and 20 were the same, but instead of 400, they used 360 in the third place value, and 360 x 20—which is 7,200—in the fourth place.
Why? Probably to take advantage of the fact that 360 corresponded more closely to the number of days in the Mayan astronomical year, which was 360 days long, with an additional five days they considered “unlucky.” (For more on the Maya calendar system, see here.)
If you want, go back to the groups of symbols near the middle of the left-hand page of the codex. Calculate the symbols again, but this time read the places as 1s, 20s, and 360s.
Check to see if you got these results:
Now find the differences between each successive pair of numbers as you did before.
Check to see that you got these results:
177 177 177 177 177 148
These numbers match the numbers at the bottom of the codex exactly! Five of the numbers are the same: 177. They represent six lunar months of 29 1/2 days each. The last number, 148, represents five lunar months (147 1/2 days).
Reading the Numbers
Now take some time to focus on the numbers. How can you figure out what the bars and dots mean? Suppose that one of these symbols counts the 1s. Which do you think it is—the bars or the dots?
The dots are simple and are similar to the small, round pebbles that people all over the world have used to keep count of things. Let’s start by counting them as single marks—1s.
Now look for the largest number of dots that appear together in one row.
Notice that there are never more than four dots side by side. If the dots are 1s, and there are never more than four dots together, how would you represent five items?
What about the bars? There are never more than three bars in a group. Could each one be a 5? A 10? Here’s a way to explore that question: Assume the bars are 10s. Now try writing the numbers from 1 to 20. (The dots are 1s, and you can use only four dots together.)
You probably discovered that if a dot is 1 and a bar is 10, there’s no way to write the numbers from 5 to 9 or 15 to 19. But if the bars are 5, you can write all the numbers from 1 to 20.
What Base Are We In?
Now that you can read individual numbers on the codex pages, look for the largest single grouping of bars and dots you can find. (A group is a single row of dots and the bars under it. Some of the numbers may be only dots or only bars.)
In some places, the number symbols run together, making them hard to read. If you look at the clearest number groups, you’ll find that the largest one contains three bars and four dots. If the bars are 5s and the dots are 1s, what is the value of this largest number represented by this group of bars and dots? Does this give you a hint about the base being used here?
The number 19—three bars plus four dots—is the largest single number in the Mayan counting system. So how could the Maya write bigger numbers?
Think about how our number system works. We use ten different symbols, 0 through 9, and combine them to write larger numbers. We can tell what each symbol means from where it appears. For example, “4” means four 1s, but “400” means four 100s, zero 10s, and zero 1s.
The same idea applies to the Maya counting system. Each group of bars and dots is an individual digit that is part of a larger number. The position of the digit tells us what its value is.
If 19 is the largest number the Maya could write in any one position, it makes sense that their system might be base twenty. To represent 20, the Maya would write one dot in the second position, just as we would write a 1 in the second position to represent 10 in our base-ten system. The next position in the Maya system would be the 400 (20 x 20) place value position.
