Explorations in Mathland: A set of texts for exploring the fanciful side of mathematics (8th grade)
Each text in the text set explores various mathematical principals from perspectives not
usually found in a standard text book. In some cases, mathematical ideas are explored through a
narrative, such as The Phantom Tollbooth and The Number Devil, for example. Other texts, like
Perfect Figures and Math Charmers, are full of ideas about numbers and their properties that a
student would likely never consider. A few texts show how Mathland is not just a place of
fantasy, but rather is the world we live in. The goal of the text set is to engage students in an
exploration of mathematical ideas, rather than the rote process of learning found in most standard
textbooks. Ideally, exploration will lead a new understanding of math’s many, many mysteries
and coincidences, and will foster a new way of thinking about numbers and math.
The text set conforms to the premise of the 8th grade SOLs, in that “Students will also
identify real-life applications of the mathematical principals they are learning that can be applied
to science and other disciplines they are studying,” and “The development of problem solving
skills should be a major goal of the mathematics program at every grade level.”1 All of the
selections touch on multiple specific SOLs, though there are no specific SOLs that are covered in
each of the texts. The goal of the text set is not to teach one area of mathematics, per se, rather
to engage students in mathematical thought. To that end, the text set also covers goals of the
Learning Principal as outlined in the NCTM Principles and Standards for School Mathematics:
“Students’ understanding of mathematical ideas can be built throughout their school years if they
actively engage in tasks and experiences designed to deepen and connect their knowledge.”2
Learning with understanding first requires students to be drawn into the subject, then to actively
reflect on the topic.
The reading grade levels range from 4.8 to 10.1 to best suit the reader. Furthermore, the
mathematical concepts range from somewhat simple ideas in number theory to rather involved
discussions on topology and probability. Because the complexity of the mathematical ideas is
independent of the reading grade level, there is a text to support a student who has lesser reading
capabilities but works well with mathematical ideas, as well as an advanced reader who struggles
with math. Below is a chart of each text’s reading grade level related to the relative complexity
of mathematical concepts and explanations.
ND: The Number Devil
A: Alice in Puzzle-Land
MDS: Math Doesn’t Suck
PT: Phantom Tollbooth
CA: Conned Again, Watson!
SLA: Secrets, Lies, and Algebra
PF: Perfect Figures
MC: Math Charmers
W: Why Do Busses Come in Threes?
MT: Math Trek
.
1
Virginia Department of Education (2001). Standards of Learning: Mathematics, 8th Grade. Richmond, VA:
VDOE. Retrieved September 2008 from www.doe.virginia.gov.
2
National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston,
VA: NCTM. Retrieved September 2008 from www.nctm.org.
The Number Devil by Hans Magnus Enzensberger (trans. By Michael Henry Heim)
Readability Level: 4.8 (Flesh-Kincaid). Reading Ease: 82.9
This story is about a boy, Robert, who meets the Number Devil in his dreams on 12
consecutive nights. Each night the Devil shares with him a new mathematical idea. There are a
large number of concepts discussed (all of which are conveniently itemized in an index at the end
of the book) which will help the student develop number sense, pattern recognition, and problem
solving, to name a few. There are many pictures and graphs which aid in developing the
mathematical ideas.
The narrative nature of the story should engage most readers, even those who are less
interested in math. Each night a new adventure unfolds in a problem-based style. The text itself
is written at a lower grade-level which makes it accessible to weaker readers. Robert seems to be
a compelling protagonist, whom readers should relate to; and the Number Devil is an intriguing
character.
I would expect most 8th graders will be able to read this on their own. The book could be
used effectively in conjunction with a lesson on patterns. Students could be assigned to read a
chapter which the class could discuss in groups or as a whole the next day.
For a relatively easy read, The Number Devil explores many very interesting properties of
numbers and patterns. The math content is appropriate for a middle school student. There is not
a great deal of new vocabulary, though mathematical terms are repeated and the explanation of
their meaning is explored through the story. There are many illustrations to help the reader
understand the math and visualize the patterns. Unfortunately, the English (as in the United
Kingdom) translation leads to a couple terms that are not used in the States: rutabaga instead of
(square) root, and prima donna as opposed to prime. A simple explanation should help the
readers while also opening a discussion on different languages’ roles in mathematics.
Alice in Puzzle-Land by Raymond Smullyan
Readability Level: 5.4 (Flesh-Kincaid). Reading Ease: 79.8
Using characters and plots from Lewis Carroll’s Alice in Wonderland, this text presents
88 puzzles of logic and mathematics. Many problems require a working knowledge of algebra
(or perhaps making a chart of input and output numbers), though many rely on logic or number
sense. The narration and underlying plot hide the fact that most of the puzzles are word
problems. Students should find the puzzles challenging, yet entertaining, and all the while
practicing their problem-solving and approaching problems from different angles.
Those students who enjoy puzzles or engaging in complex thinking will definitely enjoy
this book. I hope that the other students (likely the majority) will find the characters engaging
enough to entertain their riddles. Perhaps the students who don’t enjoy math very much will
enjoy the logic puzzles and find their solving to be enjoyable. Thinking logically may better
prepare them for further mathematical thinking. The varying levels of difficulty of the puzzles
provide something for all students to ponder.
This text will likely be used to augment a standard lesson. Particular problems would
lend themselves well to a lesson in algebra, for example. They could be used as warm-up
problems or homework problems, as most students will not have any difficulty reading and
understanding them. Group or class discussions could occur on how different people solved the
puzzles in different ways.
The text itself is very accessible to most 8th graders. More importantly, the math
involved is not too difficult, but provides a good base for thought and discussion. There is very
little, if any, explicit discussion of mathematics outside of speaking about numbers, although,
again, a working knowledge of algebra is helpful to many puzzles. In terms of thinking
mathematically and especially interpreting word problems, this book ranks among the top in the
text set.
Math Doesn’t Suck by Danica McKellar
Readability Level: 6.8 (Flesh-Kincaid). Reading Ease: 73.3
This text is specifically written for middle school aged girls. The author herself struggled
in math classes until sometime in middle school when it “clicked”. She went on to major in
math in college and subsequently wrote this book to help other young ladies. The text is
presented like a magazine targeting the same age group (Seventeen, for example). The math she
explains follows closely to most middle school math curriculums.
Girls will at least be drawn to the user-friendly format of the text. The mathematics can
be dry at sometimes, but the explanations are always conscientious of the struggling reader.
There are also intermittent “quizzes” to break any monotony (“Are you a math-o-phobe? Take
this quiz!”) Most girls will also relate to the author and the girl-specific writing style. My guess
is that some girls might be over the super-cutesy girly thing, but perhaps not as much in middle
school as in high school. Fans of Hannah Montana and Hillary Duff will also become fans of
Danica McKellar.
I would hesitate to assign reading in this book to every girl in the class. Although the
mathematical ideas are well-developed, I get a hint that McKellar is pandering to her audience.
On the other hand, there will be a handful of girls in each class who will be completely receptive
to this unique approach to math. In that case I might recommend readings that enforce that day’s
lesson to individual students. This text might help illuminate some key ideas by being further
inside the girls’ Zone of Proximal Development.
The basis of this text is that it is actually a text book, though it makes every possible
effort to not appear to be one. The math covered is specific to the middle grades and it is written
on exactly that grade level of readability. New vocabulary is in bold, then defined in multiple
ways, all geared towards the reader. Concepts are thoroughly developed and integrated into later
chapters.
The Phantom Tollbooth by Norton Juster
Readability Level: 6.9 (Flesh-Kincaid). Reading Ease: 74.8
In this classic, whimsical novel, Milo travels to Dictionopolis and Digitopolis and meets
many wondrous inhabitants. Milo meets the Dodecahedron, the Mathemagician, and .58 of a
child, among others. The mathematical ideas are presented so originally that readers cannot help
but to be engaged in the number-speak and explanations of infinity and averages.
This fantastic narrative can be enjoyed without thinking at all about the mathematics, so
I’m sure even the less inclined will still agree to read this book. The value of the reading will
come by a discussion about the mathematical ramifications. Students who merely read the text
with minimal thought paid to the mathematics can still be brought into a group or class
discussion. In some cases, students may not realize the impact of the math until a later
discussion. Every student should enjoy the escapism of the story, and those who do not
independently reflect on the math will do so in a following class.
The Phantom Tollbooth is written very close to an 8th grade level, so it is likely many
students will need some structure to realize the full benefits. The text size is inviting and there
are pictures throughout illustrating some of the characters and their mathematical forms.
However, by giving a framework before reading (introduce the characters, give a basic plot
summary, talk about the nature of the story) and discussing the principals as a class and in groups
after reading, students will fully feel the impact of the story. I would definitely also structure
activities around the mathematics that are introduced. For example, the discussion with the .58
of a child lends itself well to an activity on exploring how averages (as well as median and
mode) can be false truths in a “sticky” set of data. A cross-curricular lesson with an English
class (in which students would read the Dictionopolis half of the book) would be an effective
way for students to read the whole book.
The longevity of this novel, I think, is a testament to its usefulness. There is not a great
deal of mathematics inherent to the story, but again, its usefulness would be in class discussions.
In that manner, this book would act as a springboard into a lesson in which I control the
vocabulary and the class constructs the meaning.
Conned Again, Watson! By Colin Bruce
Readability Level: 7.7 (Flesh-Kincaid). Reading Ease: 67.0
Conned Again is based on the mysteries of Sherlock Holmes and his assistant, Dr.
Watson. In most of these cases, Watson is confronted with mysteries which Holmes handily
solves using probability theory. The computational math involved gets quite intricate (I
discussed many of these cases in college/grad-level probability, statistics, and combinatorics
courses) however the principals (theoretical versus actual probabilities and counter-intuitive
problems, to name two) can be boiled down and easily understood by 8th graders.
Readers at all levels will enjoy the mysteries, often involving a bit of vice, and Holmes’s
succinct explanations. Thankfully, the book glazes over the computational mathematics enough
to hold anyone’s interest, though leaves enough in for some poignant insights to be gleaned.
Students should enjoy trying to stay one step ahead of Watson and anticipate how Holmes will
sum it all up.
Given the reading level of this book, I would expect most students to be able to read it
with a general understanding. Principally, the mathematics involved will need some preparatory
statements before students read and discuss. For example, before assigning the reading, I can tell
students that this case involves flipping a coin. We could discuss the theoretical probability
versus an experienced probability and how and why they might differ. After reading, students
can be guided in a discussion on the particulars of the case. I would probably assign one chapter
at a time throughout a lesson on probability and statistics, and would probably skip some
chapters or assign them only for enrichment purposes (like the mystery based on the “Monty
Hall” problem – an extremely counter-intuitive problem involving three mystery doors.)
New vocabulary and concepts will only be made apparent to most 8th graders after
several re-reads or a discussion about the problems. The text itself will not be a challenge to
most students, but the mathematics can get sticky. There are many illustrations that will help
readers understand the case. Many illustrations are in the form of charts or graphs which will
also enhance students’ interpretation skills.
Do the Math: Secrets, Lies, and Algebra by Wendy Lichtman
Readability Level: 8.4 (Flesh-Kincaid). Reading Ease 70.4
This teenage drama follows the daily routine of 8th-grader Tess and her mathematical
observations about the world around her. She puts many of her ideas in the form of equations or
inequalities, Venn diagrams, and geometrical representations. This is an original book that
shows the power of abstract and algebraic thinking.
This book is primarily written for girls, given that there is much discussion of boys,
gossip, and other 8th grade “girly” things. The story is intriguing, with several inter-weaving plot
lines and mysteries, that it could stand alone without the mathematics (though it would lose some
originality points). The math involved is not extremely complicated. It may help some less
mathematically inclined students to understand how variables work (in place of one number, or
person, or many people in some cases) and other mathematical ideas including percentages, the
quadratic equation, and solving equations (what you do to one side, you must do to the other).
Any girl, probably, will enjoy this text, though the mathletes in the class may find the math a
little less than stimulating.
I wouldn’t directly assign this book to an entire class, obviously, since some boys might
be uninterested (or feign so). However, for any kind of book report or enrichment reading this
book would be great. An interesting extension may be to do have students write their own essay
using their current math vocab to describe events going on in their lives. In that way, it is similar
to my toolkit item where students make up a story to describe a graph. Although this book is at
grade level, I think girls who are reading a grade or two below level may be able to hang on with
little support because the story is so enthralling. In Henrico Co middle schools, there are reading
periods throughout the week. This would be a perfect book to (just happen to) have lying around
the classroom.
The mathematics are made so reader friendly in this text that it is actually humorous.
Students will learn vocabulary through context clues. (For example, Tess draws inequalities
between people based on their moral and physical characteristics.) The way mathematical ideas
are applied throughout the book is charming and original, with no need to formally define any
concepts. Chapter titles, such as “DNE”, “The Number Line”, and “The Additive Peroperty of
Equality” give the reader a heads up on what is to come.
Perfect Figures by Bunny Crumpacker
Readability Level: 8.4 (Flesh-Kincaid). Reading Ease: 65.3
This text explores numbers (0-12, 100, one-thousand, one-million, and infinity) like no
text I know of has ever done. The author looks at numbers through so many lenses: mythology,
literary, historically, religiously, magically, cryptically, conspiratorially, and on, and on. It is
truly a fascinating adventure through integers we generally take for granted.
This book can be enjoyed on so many levels; however, much would be lost on the
average 8th grader. Therefore, I would likely choose passages to read aloud to the class. Or, as a
project, I could give one section or chapter to groups to really dissect and present on. That may
be the best way for students to figure out what is most meaningful to them, and then present that
to the class.
Students ought to find interest in some aspect of each number: there are so many ideas
presented. Furthermore, there is not a great deal of mathematics involved, just wonderment
surrounded these numbers. On a first reading, there is little for students to be hung-up on: they
will likely glaze over pieces they find inconsequential. The great thing about this book, though,
is that there are so, so many ideas that you can forget ten things the author says about each
number and still walk away with a vastly different understanding than you had before.
Hopefully students will find bits and pieces to be beyond intriguing.
The value of this text is that you can open it to any page and become enveloped by the
ideas. Reading on you may find nothing at all of interest in the following paragraph. This
feature allows students to pick and choose new ideas to learn and others to let go. The style of
the writing also encourages this by giving many, many rapid-fire examples and anecdotes. The
size of the text and less-than-inspired illustrations may not keep as many students interested
unless they know that they will not be required to remember everything in the chapter.
Math Charmers by Alfred s. Posamentier
Readability Level: 8.6 (Flesh-Kincaid). Reading Ease 59.7
Math Charmers is an entire collection of weird, clever, and surprising mathematical
anecdotes. The author does much “toying” with numbers to come up with his charmers.
Anyone who has ever seen a fun mathematical coincidence will find something fun to
think about in this book. (Try punching 1/7, 2/7 3/7 4/7 etc in a calculator and check out the
repeating decimals.) In this book, it’s the math that drives the text, which support the figures,
numbers, and charts. Less enthusiastic students would probably benefit from a teacher skimming
through to find the fun ones, rather than looking through on their own. Indeed, many of the
charmers involve some involved mathematics; however, there are dozens that will enhance any
student’s number sense, or that they will at least find intriguing.
Although many in the class could read this, the mathematics of some are quite involved,
and I wouldn’t want students to get frustrated on one section. So, I would probably find
particular chapters and read them aloud or assign them as an extension of a lesson. Or, similar to
Perfect Figures, certain chapters could be assigned to students to fully digest, then report back to
the class.
This text moves pretty quickly through the material. Little time is spent on vocabulary (it
is assumed the reader knows what a palindrome is, for example, or can figure it out from
contextual clues.) Students reading on their own may become frustrated or too intimidated to
continue. For that reason, small, digestible portions only will be assigned for reading.
Why do Buses Come in Threes? by Rob Eastaway and Jeremy Wyndham
Readability Level: 9.3 (Flesh-Kincaid). Reading Ease: 58.6
This book describes how we all live in Mathland. Through many scenarios, the authors
show that you could look at almost any situation from a mathematical perspective, and you’ll
usually come up with something good. Exploring every day life through a mathematical
perspective is indeed an adventure. Some of the mathematics delves deeply into statistics and
calculus, yet much of it is pattern recognition and good old fashioned common sense.
Students might be surprised to see math in action in so many ways. Hopefully, that will
be encouraging as they continue to read. Ideas in the text are well presented and develop at a
pace that most readers will keep up with, for most chapters. Also, there are so many topics
covered, that most students will find at least one chapter to be quite interesting.
Because this book is pure application, I would likely use it in conjunction with a standard
lesson. Also, some students will have difficulty reading and understanding it, especially as some
of the math goes beyond what they will see until high school (or later) and the advanced reading
level. Certain chapters will apply to specific lessons and may be assigned reading. Even still,
that will be after a standard lesson so the ideas are fresh in the student’s minds. Hopefully the
text will help them see how and why it works. Activities or a discussion would take place on the
text. The pictures and diagrams will aid comprehension as well.
Ideas in the text are developed completely over some time, though it does not dwell on
explaining vocabulary. Readers must figure new words out from context clues and pictures.
Sometimes the language drifts a bit into colloquialisms, which is a welcome change in any book
about math; it also puts a human voice to the text and may be a bit easier to relate to.
Math Trek by Ivars Peterson and Nancy Henderson
Readability Level: 10.1 (Flesh-Kincaid). Reading Ease: 56.4
The format of this text is very kiddy-looking, however, it is at an advanced reading level and the
mathematics is not simple stuff. It is full of interesting concepts for students: binary numbers,
pattern recognition, fractals, and some interesting topography. There are also multiple activities
that involve construction and calculations.
Reluctant learners may be drawn in by the simplistic-looking cover and layout. By the
time they discover that there is serious mathematics within it will be too late to escape! If used
effectively in conjunction with a lesson, the text will lead to fun, yet effective learning. The
activities are an interesting and fun extension of the lessons that show math in action.
I may use this text independently of the official text, though it will parallel. I might
assign students to skim through the book and find a chapter they are interested in. Then,
probably with partners, have them read and digest the chapter and pick an activity to do. They
can do the activity and present it and the underlying mathematics to the class. Although the
reading level seems high, with all the pictures and large font I don’t think students will be
intimidated or have too much trouble understanding, especially in groups of two or more. The
completion of the activity will likely be done in class so I will be there to assist comprehension.
The readability level seems high to me – there are so many pictures and graphs that can
help students. New vocabulary is bolded and defined and there are many examples provided of
each concept. The activities in each chapter help the reader to fully understand the ideas.
Bibliography
Bruce, C. (2001). Conned Again, Watson! Cautionary Tales of Logic, Math, and Probability.
Cambridge, MA: Perseus Publishing.
Crumpacker, B. (2007). Perfect Figures. New York: Thomas Dunne Books.
Eastaway, R. & Wyndham, J. (1998). Why Do Buses Come in Threes? The Hidden Mathematics
of Everyday Life. New York: John Wiley & Sons, Inc.
Enzensberger, H. M. (translation Heim, M.H.) (1997). The NumberDevil: A Mathematical
Adventure. New York: Metropolitan Books.
Juster, N. (1961). The Phantom Tollbooth. New York: Random House.
Lichtman, W. (2007). Do the Math: Secrets, Lies, and Algebra. New York: Greenwillow
Books.
McKellar, D. (2007). Math Doesn’t Suck: How to Survive Middle School Math Without Losing
Your Mind or Breaking a Nail. New York: Hudson Street Press.
Peterson, I. & Henderson, N. (2000). Math Treak: Adventures in the MathZone. New York:
John Wiley & Sons, Inc.
Posamentier, A. S. (2003). Math Charmers: Tantalizing Tidbits for the Mind. Amherst, NY:
Prometheus Books.
Smullyan, R. (1982). Alice in Puzzle-Land: A Carrollian Tale for Children under Eighty. New
York: William Morrow and Company, Inc.