The Mathematics of Alice’s Adventures in Wonderland S U S A N B. T A B E R A LICE’S ADVENTURES IN WONDERLAND, the captivating book first published in 1865, began as a story told by a young mathematics lecturer to the three Liddell sisters, ages 8 to 13, during an afternoon of rowing on the river. The story might have evaporated into the summer’s air but for Alice Liddell, then aged 10, who asked Charles Lutwidge Dodgson to write it down for her. Although Dodgson himself illustrated the first copy, which he gave to Alice as a Christmas gift in 1864, he later expanded the story, had it illustrated by the famous artist John Tenniel, and published it under the name of Lewis Carroll (Cohen 1995). Dodgson believed that much of mathematics could be taught through games, riddles, and puzzles. He wrote many books for college and high school mathematics in which he “corrected what he saw as gaps, inconsistencies and inaccuracies in texts” (Cohen 1995, p. 254), and he devoted the last years of his life to writing books about symbolic logic. He invented box diagrams and systems of trees that simplified the symbolizing of premises and was ahead of his time in the use of truth tables to solve logic problems. His humorous examples and puzzles were written to “help his readers learn without a mighty struggle” (Cohen 1995, p. 496). Alice’s Adventures in Wonderland contains a variety of mathematical themes, jokes, and puzzles that can profitably be explored by students in the SUSAN TABER, taber@rowan.edu, teaches undergraduate and graduate courses in mathematics pedagogy and curriculum at Rowan University, Glassboro, NJ 08028. She enjoys working with classroom teachers to study the development of students’ mathematical knowledge. middle grades. Some of these are multiplication of rational numbers, the distinction between multiplicative and additive change, similarity and proportionality of geometric figures, positive and negative numbers, systems for measuring and representing time, logical reasoning, and number bases other than ten. Multiplication by rational numbers less than 1 Alice changes size twelve times during her adventures. Four of the first five changes occur when she drinks a potion from a bottle or eats a cake, but she has no reliable way of predicting whether she will grow larger or smaller. Drinking from the first bottle makes her shrink to 10 inches, but drinking from the bottle she finds in the White Rabbit’s house makes her grow too large for the house. Eating the first cake enlarges her from 10 inches to over 9 feet tall, but eating the little pebble cakes shrinks her to a small enough size so that she can leave the Rabbit’s house. Alice begins to be able to control the changes in her size after her conversation with the Caterpillar. “One side will make you grow taller, and the other side will make you grow shorter.” “One side of what? The other side of what?” thought Alice to herself. “Of the mushroom,” said the Caterpillar. . . . After a while she remembered that she still held the pieces of mushroom in her hands, and she set to work very carefully, nibbling first at one and then the other, and growing sometimes taller, and sometimes shorter, until she had succeeded in bringing herself down to her usual height (Carroll 2004, p. 61). V O L . 1 1 , N O . 4 . NOVEMBER 2005 Copyright © 2005 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM. 165 Worksheet 1 Find the changes in Alice’s height after she drinks each potion. Then write an equation that represents the change. STARTING HEIGHT POTION NEW HEIGHT EQUATION 54 inches 1/9 as tall 54 ___ 1/9 = _____ 54 inches 1/3 as tall 54 ___ 1/3 = ______ 54 inches 1/6 as tall 54 inches 5/6 as tall 60 inches 1/4 as tall 60 inches 3/4 as tall 60 inches 1/3 as tall 60 inches 2/3 as tall 60 inches 3 times as tall 60 inches 4/5 as tall 18 inches 2 times as tall 18 inches 3 times as tall 18 inches 2 1/2 times as tall 18 inches 2/3 as tall 18 inches 5/9 as tall 18 inches 6 times as tall 18 inches 3 1/3 times as tall Fig. 1 Changes in Alice’s height Through experimentation, Alice discovers how to grow taller or smaller to suit her purposes—to participate in the tea-party or to unlock the door and enter the garden. Each of Alice’s changes in size can be represented mathematically as multiplication. When she grows taller, her height is multiplied by a number greater than 1; when she grows smaller, it multiplies by a number less than 1. Several studies have shown (Graeber 1993; Greer 1992; Taber 1999) that students have difficulty thinking of situations involving multiplication by a number less than 1 as multiplication. Although they can compute the 166 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL product of two fractions or decimals, when given a word problem with a fraction or decimal multiplier less than 1, they think that the result is found by dividing by the decimal or fraction. I have successfully used a discussion of Alice’s changes in size to help students extend their understanding of multiplication by whole numbers to include multiplication by fractions (Taber 2002). Having students make a list of various size changes (see fig. 1) that Alice might undergo and asking them which operation sign should be placed between the two factors provide a springboard for discussing the effect of multiplying by a number less than 1. As my stu- dents worked with a variety of examples, increasing and decreasing Alice’s height, they generalized their experiences by stating that multiplication by a number less than 1 gives a result that is smaller than the beginning quantity. I then introduced finding the product of two fractions by having students work in groups of three on worksheet 2 (fig. 2). The first question is similar to the problems they did before. The second question is designed to help them extend finding a fraction of a whole number to finding a fraction of a fractional part. Most students solved the problems by finding the product of the two numbers and by partitioning the drawing to show the result. Problem 1 in figure 3 shows how Lisa, Carla, and Ted partitioned each section into fourths and collected five of them to show Alice’s new height of 1 1/4 feet. Problem 2 in figure 3 shows how Erica, Lila, and Hilary shaded two of the 5/8 to show 2/5 of 5/8. We also explored the idea of a limit while examining the results of multiplying Alice’s height by various numbers less than 1. In chapter 2, Alice is afraid that if she keeps shrinking she will completely disappear and cease to exist. I asked students to find a fraction that would make Alice’s height become 0. Of course, no such number exists, other than 0. No matter how small Alice became, she would exist, and her height could be represented by a number greater than 0. Worksheet 2 1. Alice is 5 feet tall. Find out how tall she will be if she eats a cake that makes her 1/4 as tall. 5 feet Show two ways to find out. 4 feet 3 feet 2 feet 1 foot 2. Alice is 5/8 of an inch tall. She nibbles some of the mushroom, but it makes her shrink to 2/5 as tall. How tall is Alice now? Show two ways to find out. 1 inch 5/8 inch Additive and multiplicative change Problems 4 and 5 on worksheet 2 (fig. 2) were included to draw students’ attention to the distinction between 1/2 of 5/6 pound and 5/6 pound – 1/2 pound. Most students had difficulty distinguishing between the two problems and either multiplied both or subtracted both. Discussing how the Cheshire Cat appears and disappears provides an opportunity to distinguish additive from multiplicative changes. The Cheshire Cat appears and disappears in pieces. Sometimes only his tail can be seen; at other times, only his head or grin remains. This situation can be described mathematically as an additive, rather than multiplicative, process. The processes of contrasting and comparing the ways in which Alice and the Cheshire Cat are transformed and relating them to mathematical operations help students understand the distinction between multiplicative change and additive change. Other transformations such as the pebbles that become cakes, the baby that changes into a pig, the flamingo croquet mallets, or the hedgehog croquet balls are not mathematical. I asked students to make a list of the transformations that occur in the book and state whether or not each transformation could be represented mathematically. 3. Mike bicycled 16 miles on Friday. On Saturday, he bicycled just 2/3 as far as he did on Friday. How far did he ride his bicycle on Saturday? Show two ways to find out. 16 miles 4. There was 5/6 of a pound of candy in the cupboard. Millie took 1/2 of it to school. How much candy did Millie take to school? Show how to find the answer in two ways. 1 pound 5. There was 5/6 of a pound of candy in the cupboard. Jason took 1/2 pound of the candy to school. How much candy was left? Show how to find the answer in two ways. 1 pound Fig. 2 Fractions of fractions V O L . 1 1 , N O . 4 . NOVEMBER 2005 167 1. Alice is 5 feet tall. Find out how tall she will be if she eats a cake that makes her 1/4 as tall. 5 feet Similarity and proportional and nonproportional transformations on the Cartesian plane Show two ways to find out. Most of the time Alice’s proportions are preserved, whether she becomes larger or smaller. The picture of Alice at the beginning of chapter 2 (Carroll 2004, p. 24), however, shows a very tall and skinny Alice; her proportions have not been preserved (see fig. 4). I introduced a discussion of similarity and proportionality by asking students to describe how the Alice shown at the beginning of chapter 2 was different from other pictures of Alice throughout the book. Students used worksheet 3 (fig. 5) to explore transformations on the Cartesian plane that result in similar figures, those in which the proportions are preserved and those that result in nonsimilar or nonproportional figures. I asked students to describe what kinds of transformations yield a proportional Alice or nonproportional Alice. Students were also asked to enlarge or reduce other geometric figures like triangles or rectangles in ways that preserved and did not preserve their proportions and to represent those transformations with mathematical language and symbols, such as “I multiplied the height by three and the width by four.” Students then classified the transformed rectangles and other figures as similar or not similar to the original figures. 4 feet 5 × 1/4 = 5/4 = 1 1/4 ft. 3 feet 2 feet 1 foot 2. Alice is 5/8 of an inch tall. She nibbles some of the mushroom, but it makes her shrink to 2/5 as tall. How tall is Alice now? Show two ways to find out. 1 inch 1.) 5/8 × 2/5 = 10/40 = 1/4 5/8 inch 2.) Negative numbers Fig. 3 Students’ solutions to worksheet 2, numbers 1 and 2 ILLUSTRATION OF ALICE BY JOHN TENNIEL, FROM ALICE’S ADVENTURES IN WONDERLAND BY LEWIS CARROLL In chapter 9, Alice meets the Mock Turtle and the Gryphon and discusses school with them. Fig. 4 “Now I’m opening out like the largest telescope that ever was!” (Carroll 2004, p. 24) 168 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL “And how many hours a day did you do lessons?” said Alice, in a hurry to change the subject. “Ten hours the first day,” said the Mock Turtle: “nine the next, and so on.” “What a curious plan!” exclaimed Alice. “That’s the reason they’re called lessons,” the Gryphon remarked: “because they lessen from day to day.” This was quite a new idea to Alice, and she thought it over a little before she made her next remark. “Then the eleventh day must have been a holiday?” “Of course it was,” said the Mock Turtle. “And how did you manage on the twelfth?” Alice went on eagerly. “That’s enough about lessons,” the Gryphon interrupted in a very decided tone (Carroll 2004, p. 111). Asking students why the Gryphon did not want to consider a twelfth day of “lessons” leads to a discussion of real-life situations in which negative numbers are reasonable and make sense and situations in which negative numbers do not make pragmatic sense. Worksheet 3 1. Graph, label, and connect the following points in order. A (10, 0) E (8, 5) I (10, 7) M (12, 7) Q (13, 5) U (12, 0) B (10, 2) F (8, 6) J (10, 9) N (12, 6) R (14, 2) V (12, 2) C (9, 2) G (11, 6) K (13, 9) O (15, 6) S (13, 2) W (11, 2) D (10, 5) H (11, 7) L (13, 7) P (15, 5) T (13, 0) X (11, 0) 2. Connect point X to point A. 3. List five examples of line segments that are horizontal. ______________________________________ ______________________________________________________________________________________ 4. List five examples of vertical line segments. ________________________________________________ ______________________________________________________________________________________ 5. How are the line segments that connect C with D and Q with R different from those you listed in questions 3 and 4? ______________________________________________________________________ ______________________________________________________________________________________ 6. Graph, label, and connect each of these points in order. A' (10, 0) E' (6, 10) I' (10, 14) M' (14, 14) Q' (16, 10) U' (14, 0) B' (10, 4) F' (6, 12) J' (10, 18) N' (14, 12) R' (18, 4) V' (14, 4) C' (8, 4) G' (12, 12) K' (16, 18) O' (20, 12) S' (16, 4) W' (12, 4) D' (10, 10) H' (12, 14) L' (16, 14) P' (20, 10) T' (16, 0) X' (12, 0) 7. Connect point X' to point A'. 8. How is this figure different from the figure for question 1? ____________________________________ ______________________________________________________________________________________ 9. How is it the same as the figure for question 1? _____________________________________________ ______________________________________________________________________________________ 10. How is the figure at right different from the two figures you drew in questions 1 and 6? _______ _________________________________________ _________________________________________ 11. Draw a figure that is 3 times as tall and 3 times as wide as the one you drew for problem 1. 12. If Jackie drew a figure that was 4 times as tall, but the same width as the one you drew for problem 1, what would be the coordinates of the following points? A ______ J ______ K ______ T ______ Fig. 5 Graphing changes in Alice’s sizes V O L . 1 1 , N O . 4 . NOVEMBER 2005 169 Complete the table below by computing each product, then translating it to the base indicated. FACTORS 4×5 4×6 4×7 4×8 4×9 4 × 10 4 × 11 4 × 12 4 × 13 4 × 14 PRODUCT BASE 10 BASE 18 21 24 27 30 33 36 39 42 45 PRODUCT EXPRESSED IN INDICATED BASE 12 = (18 + 2) 13 = (21 + 3) Solution: FACTORS 4×5 4×6 4×7 4×8 4×9 4 × 10 4 × 11 4 × 12 4 × 13 4 × 14 PRODUCT BASE 10 twenty twenty-four twenty-eight thirty-two thirty-six forty forty-four forty-eight fifty-two fifty-six BASE 18 21 24 27 30 33 36 39 42 45 PRODUCT EXPRESSED IN INDICATED BASE 12 = (18 + 2) 13 = (21 + 3) 14 = (24 + 4) 15 = (27 + 5) 16 = (30 + 6) 17 = (33 + 7) 18 = (36 + 8) 19 = (39 + 9) 1# = (42 + 10) 1@ = (45 + 11) Note: # stands for 10 units; @ stands for 11 units. Fig. 6 Alice’s “multiplication table” Systems for measuring and representing time The tea-party with the Mad Hatter and the March Hare provides an opportunity to learn about the systems that humans have invented for measuring the passage of time and the history of the calendar. Watches and calendars are useful, because everyone agrees on conventions for their use. The conventions followed by the Mad Hatter and the March Hare are clearly not those followed by Alice or by us. For example, the Hatter’s watch indicates the day of the month instead of the hours of the day, and the Hatter tells Alice that it is always 6:00, or tea-time. Carroll is also making fun of a situation that existed in Great Britain and all its colonies until September 14, 1752. Both the hat (Carroll 2004, p. 80) and the watch of the Mad Hatter indicate that the 170 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL date is 10/6, October 6 (Dreyer 1981). When Alice says that it is the fourth, the Hatter sighs, “Two days wrong.” In fact, the date October 6, 1582, did not exist for most Europeans. In 730, Saint Bede the Venerable announced that the Julian calendar, adopted in 46 B.C., was eleven minutes fourteen seconds too long, which resulted in a cumulative error of about one day for every 128 years. To correct the accumulated errors, Pope Gregory decreed that the day following October 4, 1582, should be called October 15, thus dropping ten days. Furthermore, to correct the length of the year, three of every four centesimal years (ending in 00) would not be leap years. Thus, 1600 and 2000 would be leap years, but not 1700, 1800, and 1900. The Gregorian calendar was adopted at once by France, Italy, Spain, Portugal, and Luxembourg and within five years by the German Catholic states, Belgium, parts of Switzerland, the Netherlands, and Hungary. For nearly two hundred years, Great Britain, which did not recognize the authority of the Catholic Church, used a different calendar than the rest of Europe, a situation only slightly less mad than the mad tea party. The British government finally imposed the Gregorian calendar on all its possessions in 1752. September 2 of that year was followed immediately by September 14. All dates preceding September 2, 1752, were marked OS for Old Style, and all dates after September 14 were designated NS (New Style). At the same time, New Year’s Day was moved from March 25 to January 1. The Hatter’s statement, “We quarreled last March—just before he went mad, you know—” (Carroll 2004, p. 82) is probably a reference to the change of New Year’s Day. Understanding the references to time and the history of the calendar in this chapter makes the chapter easier to understand and also provides an opportunity for students to learn about related topics such as other calendars and time-keeping systems, the origins and reasons for time zones, and to understand that our time-keeping systems are human inventions imposed on the natural world. Logical reasoning Alice makes a number of errors in logical reasoning during her adventures. Students enjoy making and comparing lists of these errors as they find them in the story. For example, when Alice decides that because a bottle is not marked “poison,” the contents are safe to drink, she commits the logical error of denying the antecedent (Heath 1974). When Alice attempts to measure whether she is getting larger or smaller by placing her hand on her head, she is attempting to measure a system with a tool that is part of the system itself. When Alice concludes that because the rabbit’s house is already too tight a fit for her, eating the pebble cakes will surely make her smaller, she is asserting one alternative by denying the other (Heath 1974). students in the middle grades. Using the book as the focus of an interdisciplinary language arts and mathematics unit will provide many opportunities to make connections among mathematics, literature, history, and culture. The base-ten number system and bases other than ten References In chapter 2, Alice begins trying to recite things that she has learned to reassure herself that she is still Alice. Let me see: four times five is twelve, and four times six is thirteen, and four times seven is—oh dear! I shall never get to twenty at that rate! . . . I must have been changed for Mabel! (Carroll 2004, pp. 25–26). Alice will never get to twenty with the multiplication table she has begun reciting, because each product is in a different base (Taylor 1952). Although this topic lies outside the typical middle-grades curriculum, it is an interesting enrichment activity for students who want to try it. Completing the chart in figure 6 helps students see the pattern of the products and explains why the pattern will never arrive at “20.” New symbols will have to be invented for the “10” and “11” in the last two rows of the table. Carroll, Lewis. Alice’s Adventures in Wonderland and Through the Looking Glass. Illus. by John Tenniel. New York: Barnes and Noble Classics, 2004. Cohen, Morton, N. Lewis Carroll: A Biography. New York: Alfred A. Knopf, 1995. Dreyer, Lawrence. “The Mathematical References to the Adoption of the Gregorian Calendar in Lewis Carroll’s Alice’s Adventures in Wonderland.” The Victorian Newsletter 60 (fall 1981): 24–26. Graeber, Anna O. “Misconceptions about Multiplication and Division.” Arithmetic Teacher 40 (March 1993): 408–11. Greer, Brian. “Multiplication and Division as Models of Situations.” In Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics, edited by Douglas Grouws, pp. 276–95. New York: Macmillan Publishing Co., 1992. Heath, Peter. The Philosopher’s Alice. New York: St. Martin’s Press, 1974. Taylor, A. L. The White Knight. Edinburgh: Oliver & Boyd, 1952. Taber, Susan B. “Understanding Multiplication with Fractions: An Analysis of Problem Features and Student Strategies.” Focus on Learning Problems in Mathematics 21 (spring 1999): 1–27. ———. “Go Ask Alice about Multiplication of Fractions.” In Making Sense of Fractions, Proportions, and Ratios, 2002 Yearbook of the National Council of Teachers of Mathematics (NCTM), edited by Bonnie Litwiller, pp. 61–71. Reston, VA: NCTM, 2002. For more articles relating mathematics and literature, see the April 2005 focus issue of MTMS, titled “Connecting Mathematics and Literature in the Middle Grades.”—Ed. Carroll wanted to “help his readers learn without a mighty struggle” (Cohen 1995, p. 496) It is not surprising that Alice’s Adventures in Wonderland is stocked with mathematical references, puzzles, and themes; after all, Lewis Carroll the author and Charles Dodgson the mathematician were the same person. It is easy to imagine Dodgson discussing the mathematical concepts and topics with the Liddell girls that he incorporated into his story. Each topic is appropriate for investigation by many V O L . 1 1 , N O . 4 . NOVEMBER 2005 171