What is the math moral of the story?
Using children's literature to teach mathematics is an approach currently being espoused by the curriculum reform movement. Publications aimed at classroom teachers regularly print articles that suggest ways to implement this approach. The National Council of Teachers of Mathematics journal, Teaching Children Mathematics, has a regular feature called "Links to Literature." These articles primarily use children's literature to provide an engaging context for problem solving.
Focusing on the "math moral" of a story is a different approach to using children's literature in the classroom. With this approach, one steps back from the details and tries to abstract a general principle. By encouraging children to look for, discover, discuss and extend math morals, teachers can help them understand important mathematical principles and gain experience in generalization and abstraction, which are important higher order thinking skills.
To see the difference in approach, consider The Doorbell Rang (Hutchins, 1986). In this story, two children share a dozen cookies equally, therefore they each have six. The doorbell rings and another child arrives, so now each child has four cookies. Again and again the doorbell rings and the children have fewer cookies each, until the children each have only one cookie. Luckily, the doorbell rings one more time and it is Grandma with more cookies. Many books and articles on integrating mathematics and literature recommend using this story.
Related activities suggested include having students: role-play the story, write their own stories, and work out in different ways how many cookies each child would have for various different numbers of children and cookies. Depending on the activity, children might need to use fractions. Some activities involve using manipulatives representing cookies to solve multiplication and division problems. (See Abrahms, 1992, pp. 40-41; Braddon, Hall & Taylor, 1993, pp. 47-48; Garcia, 1994, pp. 16-17; Griffiths & Clyne, 1988, pp. 26-27; Welchman-Tischler, 1992, pp. 50-51; Whitin & Wilde, 1992, pp. 16-17.) Such activities use the story as an appealing and meaningful context in which students can develop skills and an understanding of the concept and the process of division, and perhaps fractions. Some authors suggest having students write their own similar stories - encouraging students both to be creative and to communicate about mathematics.
What about a math moral? What general mathematical statement can be made about what happens when more children arrive? As the same number of cookies is divided equally among more and more children, each child gets fewer cookies. Therefore, the math moral is: If you divide a given set of objects equally into more and more subsets, each subset contains fewer objects.
When Grandma arrives with more cookies, each child would now get more cookies. This leads to another general principle. If we have two sets, one with more objects than the other, and divide each evenly into the same number of subsets there will be more in each subset of the bigger set. Teachers should not pass up the opportunity to discuss these general principles.
Some examples of other stories with math morals follow.
Example 1. Incomplete information can be inaccurate. A classic folk tale from India illustrates the dangers of incomplete information. A group of blind men encounter an elephant; each touches only one part of the elephant and thus receives a misleading idea of elephants. For example, one touches the trunk and concludes that an elephant is like a snake. Understanding the dangers of incomplete information is especially relevant for the field of mathematics. Forming conjectures and recognizing patterns are important, but the ultimate goal of mathematics is to support these conjectures with formal proof.
Two versions of this story currently in print are Karen Backstein's The Blind Men and the Elephant (1992) and Ed Young's Seven Blind Mice (1992). In Young's picture book version, the blind men are replaced by strikingly colored blind mice. In this version, the 7th blind mouse is able to ascertain the true nature of the elephant, without outside assistance, by running all over the elephant. Thus, this version has a moral in the usual sense, that persistence can bring increased knowledge. This moral is relevant to the study of mathematics. Too often, students do not succeed in mathematics because they give up too quickly.
Pattern recognition is an important mathematical skill. Students are often told to find or to continue the pattern. You know when you have found the pattern when you find a rule that generates the pattern. However, there is a problem here that is not often emphasized with young children. Different patterns may begin in the same manner. For example, many patterns begin "1, 2, 4. . . ." One possibility is "1, 2, 4, 8, 16, 32, 64 . . .," in which each number is double the previous number. Another possibility is "1, 2, 4, 7, 11, 16, 22 . . .," which is generated by starting with the number one, adding one to get the next number, then adding two to the second number to get the third, three to the third number to get the fourth, etc. We cannot determine how a pattern will continue by merely looking at the first three or even the first thousand terms. If we emphasize this concept early with children, then when older, they may better understand that a set of specific examples is not a proof.
Example 2. Units matter, standard units are necessary.
Students often forget or are sloppy with units when measuring. Many students simply do not realize the importance of using correct units. How Big Is a Foot (Myller, 1972) is written specifically to illustrate this math moral. A king orders one of his craftsmen to make a bed for the queen. The king measures how big it should be by stepping around the queen and gives the measurement in feet (his own). The finished bed is too small because the craftsman used his own foot as a measure. Thus, the story emphasizes the importance of using a uniform standard of measurement.
Example 3. Division is not commutative.
A mathematical operation * is commutative if a * b = b * a for all a and b. Addition and multiplication are commutative, division and subtraction are not. Some children optimistically believe that all operations are commutative, since cummutative operations are a bit simpler - less care is needed in reading or writing the operation. Other students may simply be confused by the symbols. They may know that there is a difference between 2 [divided by] 8 and 8 [divided by] 2, but they are not sure just what it is.
Nine-in-One Grr! Grr! (Xiong & Spagnoli, 1989) is a Hmong folk tale in which a tiger is unhappy because she has no children and asks the god Shao for help. Shao tells the tiger that she will have nine cubs in one year as long as she can remember the words nine-in-one (nine cubs per year, which is equivalent to 9 [divided by] 1 cubs per year). Raven, worried about a tiger population explosion, convinces the tiger that the words are one-in-nine (one cub per nine years, which is equivalent to 1 [divided by] 9 cubs per year). The difference can be emphasized by asking the children to figure out how many tiger cubs a single pair of tigers would produce in nine years under the two plans.
How the Ox Star Fell from Heaven (Hong, 1991) is a Chinese story with a similar moral. The story begins with all the oxen living the good life in heaven while people toil and starve on the earth below. Sometimes, the people went eight days without eating. The Emperor of Heaven takes pity on the people and decrees that they should eat at least once every three days. The Ox Star is sent to earth to deliver the message, but he mistakenly says that they shall eat at least three times in one day. This is good for people, but the oxen are punished for the Ox Star's mistake.
Example 4. Exponential growth is very rapid.
Start with the number 1 and double it to get 2, double again to get 4, double again to get 8. While these are not large numbers, if you continue doubling the numbers they get very large fairly quickly. By the 32nd doubling, for example, you would have the enormous number 4,294,967,296. The King's Chessboard (Birch, 1988) is a good illustration of this concept. A foolish and proud king offers a wise man a reward and allows him to name his own reward. The wise man asks for one grain of rice the first day, then two the second, with the amount of rice doubling each day for 64 days, one for each square on a chessboard. The king realizes on the 32nd day that it is impossible to deliver the reward and learns a lesson about humility. Other books that explore the effects of repeated doubling are A Grain of Rice (Pittman, 1986) (another version of the same story set in China), Melisande (Nesbit, 1989) and Two of Everything (Hong, 1991).
To generalize the math moral, if you start with any number greater than 0 and repeatedly multiply by any number bigger than 1 there will be rapid growth. (If you were to multiply repeatedly by a number greater than 0 but less than 1, the number would shrink away to 0 rapidly.) Students could investigate and then write their own stories about what happens when they keep cutting something in half. Encouraging children to look for general principles, "math morals," in stories, rather than merely using the literature as a context for problems and activities can help children build reasoning skills and makes mathematics more meaningful.
Abrahms, A. (1992). Literature-based math activities: An integrated approach. New York: Scholastic.
Braddon, K. L., Hall, N.J., & Taylor, D. (1993). Mathematics through children's literature: Making the NCTM standards come alive. Englewood, CO: Teachers Ideas Press.
Garcia, A. (1994). Math and literature: Hands-on activities for 35 literature titles. Cypress, CA: Creative Teaching Press.
Griffiths, R., & Clyne, M. (1988). Books you can count on: Linking mathematics and literature. Portsmouth, NH: Heinemann.
Welchman-Tischler, R. (1992). How to use children's literature to teach mathematics. Reston, VA: National Council of Teachers of Mathematics.
Whitin, D. J., & Wilde, S. (1992). Read any good math lately? Portsmouth, NH: Heinemann.
Backstein, K. (1992). The blind men and the elephant. New York: Scholastic.
Birch, D. (1988). The king's chessboard. New York: Dial.
Hong, L. T. (1991). How the ox star fell from heaven. Morton Grove, IL: Albert Whitman.
Hong, L. T. (1991). Two of everything. Morton Grove, IL: Albert Whitman.
Hutchins, P. (1986). The doorbell rang. New York: Greenwillow.
Myller, R. (1972). How big is a foot? New York: Dell.
Nesbit, E. (1989). Melisande. San Diego, CA: Harcourt Brace.
Pittman, H. C. (1986). A grain of rice. New York: Hastings.
Xiong, B., & Spagnoli, C. (1989). Nine-in-one Grr! Grr! San Francisco, CA: Children's Book Press.
Young, E. (1992). Seven blind mice. New York: Philomel Books.
Jane E. Friedman is Associate Professor, Mathematics and Computer Science Department, University of San Diego, San Diego, California.