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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
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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
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