Honors Thesis

Bridging the Gap: Integrating Literature into

Mathematics Education

Scott Davis

Thesis Advisor: Dr. Ann Ciasullo

The word technology comes from the Greek term “techne,” which at the time had a

nearly equivalent meaning to the Latin word “ars;” the root from which the word “art” stems. As

can be inferred from the modern words constructed from these roots, the Greek and Latin origins

encompassed many more activities than most people would have initially thought. Carpentry,

medicine, sculpture, even rhetoric was included in these two phrases. The ancient and medieval

societies made no distinction between what we would most likely consider sciences and the fine

arts.1 Only with the Enlightenment, and the evolution of scientific thinking, did the world see a

true breaking between the “sciences,” and the “arts.”

As time passed, and the gap between “scientific” and “creative” ways of thinking

continued to widen, people too began to prefer one method to the other.2 This distinction,

unfortunately, has become incredibly commonplace in the modern United States where people

tend to align themselves with either the arts or the sciences, with few pursuing a combination of

the two. Everyone has heard of the left-brain/right-brain distinction, and though the scientific

validity of this difference is not entirely accurate, people still often distinguish or identify

themselves as one or the other. What this means, then, is before even given the task of solving a

math problem or interpreting a short poem, many people will already have assumed a

predisposition that they will either enjoy and succeed with the “task,” or instead will struggle

through it and feel no joy when they are done. This assumed mindset can alter people’s

experience with the subject matter before they even interact with it.

Research has shown, in fact, that perhaps the distinction between left-brain and right-

brain could be the wrong division completely. In a 2009 study, psychologists Lee Thompson,

Sara Hart, and Stephen Petrill found that there is actually a genetic overlap between math

problem-solving skills and reading decoding, while “math fluency,” a measure of timed 1 Schatzberg, Eric. 2012. “From Art to Applied Science.” Isis 103 (3) (September): 555-60. 2 Ibid

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calculation, and reading fluency also share genetic overlap.3 The skills within each of these

separate areas (reading and math) were completely independent of one another though. This

means that while our brains don’t really distinguish between the actual subject matter, they do

manage to detect the different types of thinking required by a certain aspect of each subject;

problem solving vs. calculating, for example.

Yet, the original left-brain, right-brain distinction remains popular within society, and

with very different connotations. In the United States, nearly all people would look down on

someone who is illiterate; it would be nearly impossible to find a job, or even someone who

sympathized whole-heartedly with that struggle. We take the ability to read for granted.

Mathematics, on the other hand, is more often viewed as a benefit; a skill that advantageous to

possess, but not wholly necessary. Unfortunately, our culture has widely accepted this belief.

While lacking the ability to read could harm one’s societal status, we are all too tolerant of

someone who “just can’t do numbers.”4 This cultural acceptance of the divide not only highlights

its presence, but continually widens the gap.

Signs of this attitude have permeated nearly all elements of our culture. In books and

movies, an understanding of mathematics is a surefire way to depict a character with extensive

intelligence.5 From Good Will Hunting to The Girl with the Dragon Tattoo, the brilliant

characters are assumed to be genius because they are portrayed interacting with higher level

mathematics. And this is by no means a new rhetorical technique. Take, for example, this

description from a popular 19th century novel: “He is a man of good birth and excellent

education, endowed by nature with a phenomenal mathematical faculty. At the age of twenty-one

3 Hart, Sara A., Stephen A. Petrill, Lee A. Thompson, and Robert Plomin. 2009. “The ABCs of Math: A Genetic Analysis of Mathematics and Its Links with Reading Ability and General Cognitive Ability.” Journal of Educational Psychology 101 (2): 388-402. 4 Rochman, Bonnie. 2013. Beyond counting sheep. Time 181 (7) (02/25): 52. 5 Fowler, David. 2010. “Mathematics in Science Fiction: Mathematics as Science Fiction.” World Literature Today 84 (3) (May): 48-52

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he wrote a treatise upon the binomial theorem, which has had a European vogue.”6 This

description of Dr. Moriarty, taken from one of Arthur Conan Doyle’s Sherlock Holmes stories,

illustrates this point perfectly.

The success of this association between brilliance and math competency relies on the fact

that society unanimously agrees that mathematics, plain and simple, is just a hard subject to

learn. We separate our ability to perform even basic math from our ability to read and

comprehend a text, no matter how difficult. Odds are, most “right-brained” people would prefer

to read Moby Dick over spending an equivalent amount of time solving basic arithmetic

problems. This preference is a direct result of the cultural belief that math is inherently difficult.

Even Barbie herself, at one time, used to say, “math class is tough.”7 This societal mindset and

view of mathematics does absolutely nothing to help improve mathematical skills in our youth.

In fact, it does the opposite by fostering an abundance of math anxiety in students, very often

leading to an avoidance of math. Even in the data filled, technological world we live in, our

education system seems to continually be unsuccessful in teaching students numeracy, the math

equivalent to literacy.8

Some people might argue that students’ math-avoidance is due to a lack of intelligence;

that maybe math anxiety is merely regular anxiety for “unintelligent” individuals attempting to

perform mathematical tasks. This belief holds no ground, however, as studies have indicated that

math anxiety has almost no correlation with scores on an IQ test. In fact, there have even been

instances of improving scores on math tests after individuals went through math anxiety

treatment which involved no mathematical instruction whatsoever.9 This means that it is possible

6 Fowler “Math in Science Fiction,” 48-527 Ashcraft, Mark H. 2002. “Math Anxiety: Personal, Educational, and Cognitive Consequences.” Current Directions in Psychological Science 11 (5) (Oct.): 181-5. 8 Ibid 9 Ibid

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to help students with math anxiety perform better in test environments, letting their true

mathematical competency show through.

A more probable reason for math-avoidance, paired with math anxiety, comes from the

fact that many students in our modern world cannot see the benefit of obtaining a math

education. But who can blame them? We have calculators, and other online resources, that

essentially do the math for us. Why, then, should we spend time learning the material, if we will

never need to perform these operations for ourselves? This precious time could instead be spent

learning other valuable skills, such as reading, which most people perform every day.

Unfortunately, kids with this belief are unable to see the greater depth that the subject has to

offer. There are far more important problem-solving skills learned through the study of

mathematics that go way beyond addition and subtraction. Without any exposure to these skills,

however, students will begin spending more time on other subjects, leading to a snowball effect

with math anxiety; students who avoid math tend to become more math anxious, leading to more

avoidance, and more anxiety.10

While there have been successful means of treating math anxiety, as previously

mentioned, it would certainly be more beneficial to avoid it altogether. A major source of this

anxiety, and one explanation for the cultural views of math, originates from the manner in which

the subject is taught. Math classes too often spend a majority of the time emphasizing aptitude

and answers, as opposed to processes and effort. This unnecessary pressure to be “correct” all the

time tends to push students with math anxiety away from math-based majors in college, and then

careers later in life.11 The moment students begin aligning themselves with the arts rather than the

sciences, many of them already write themselves off as not being math-oriented people, affecting

10 Ashcraft, “Math Anxiety,” 181-5.11 Ibid

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their eventual life choices for the wrong reasons.12 Another, more obscure explanation for

student’s disinterest, coming from Ellen and Bob Kaplan, the founders of the Math Circle in

Boston, comes from the belief that math is a subject which can be taught at all.13 The Kaplans

believe that mathematics is best learned through self-discovery and a construction the students

help to form, as opposed to information merely being handed to them. If more teachers began

taking this approach, the Kaplans claim students would begin to experience the aspects of math

that most instruction leaves out: “creativity, playfulness, wonder, and boundless curiosity.”14

While these are by no means the only possibilities, in either case, when instructors

demand perfection and place an importance on obtaining correct answers rather than giving

support for errors and mistakes, there tends to be a much higher rate of math avoidance in the

students after the course.15 Similarly, the rigid structure of most math courses limits the exposure

students have to the field. Greg Tang, an author of children’s math stories including “The Grapes

of Math,” has noted that in his experience nearly all younger students claim to enjoy math, but

this love begins to dwindle as they get older.16 One possible reason for this is because, unlike

reading, most students first encounter math in a classroom environment, meaning that the subject

immediately becomes compulsory as opposed to an activity for fun.17 As students’ interest in

school begins to decrease, so too will their interest in math. Another possibility, proposed by

Tang, again places the blame on the means of instruction. He believes most schools remove the

problem-solving aspect of math and instead present the material as lists of computations and

formulas. As he puts it, “math quickly becomes a jumble of rote methods and mechanical

12 Ashcraft, “Math Anxiety,” 181-5.13 Kennedy, Steve. 2003. “The Math Circle.” Math Horizons 10 (4) (April): 9-1014 Ibid15 Ashcraft, “Math Anxiety,” 181-5.16 Tang, Greg. 2002. “Taking the WORRY Out of MATH.” Book Links 12 (2) (Oct): 44-45. 17 Rochman, Bonnie. 2013. “Beyond Counting Sheep.” Time 181 (7) (02/25): 52-54

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procedures with little understanding or intuition.”18 Even the American award-winning novelist

David Foster Wallace agreed with this sentiment. In a review of two “math-melodramas” (to be

discussed later), he pointed out the problem with math education is that students barely skim the

surface of the subject, judging the whole field of mathematics on the introductory material,

“which is roughly analogous to halting one’s study of poetry at the level of grammar and

syntax.”19 Exposure to the more abstract and pure aspects of mathematics requires deeper

thinking, and much more problem solving creativity than calculator math necessitates.

Moreover, research has shown that most of society doesn’t even know what

mathematicians actually do; the general public’s understanding of the profession greatly differs

from the reality of professional mathematicians. This misconception holds especially true in

younger students, and does not occur solely in the United States. In a study of 12-13 year olds’

perceptions of mathematicians, across five countries, researchers found that the connection

between the student’s understandings and the actual work that mathematicians perform is nearly

invisible.20 The blame, as Tang, Rochman and Wallace see it, should be placed on the means of

instruction. Since students’ exposure to mathematics is primarily dependent on their interaction

with it in school, when instructors present the material in a cut-and-dry manner, students cannot

imagine the subject in a more interesting light. They begin viewing mathematicians in the

stereotypical manner, because they have nothing else to base their opinion on.

This constructed image, according to the study, includes many negative elements such as

nerdy glasses and antisocial behavior. Similarly, these 12-13 year olds think that

mathematicians’ job involves nothing more than long tedious calculations – essentially an

18 Tang, “WORRY Out of MATH,” 44-45.19 Wallace, David Foster. 2000. “Rhetoric and the Math Melodrama.” Science 290 (5500) (Dec. 22): 226320 Picker, Susan H., and John S. Berry. 2000. “Investigating Pupils' Images of Mathematicians.” Educational Studies in

Mathematics 43 (1): 65-70.

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extension/complication of the type of problems that they encounter in their own classrooms.21

These intermediate mathematical skills, though necessary to study many advanced math

concepts, actually make up very little of a mathematician’s time, but with the customary

presentation of the material, students’ struggle to picture this. Unfortunately, if this cultural view

permeates too far and too strongly into younger students, it is unlikely that very many will opt to

pursue such a negatively viewed profession, once again leading to an avoidance of the subject. If,

however, instructors can present the material in creative and original ways, students will begin to

better understand what mathematicians do, and gain a better appreciation for math in general.

In fact, in the past few decades, there have been many successful programs which have

helped students gain an appreciation of mathematics. Laura Overdeck, a Princeton-trained

astrophysicist, began her “Bedtime Math” program whose mission it is to change the way

students go to bed.22 Although most parents already read to their children before putting them to

sleep, Overdeck argues that by if parents included solving just a single math problem in this

routine, that problem-solving in mathematics becomes much more enjoyable. And better yet, it

can also spur kids’ interest in math before they even begin school. By using a calendar of fun

events, such as Cookie Monster’s birthday, Overdeck’s program has been so influential than one

customer has begun to use math as a threat; “If you don’t brush your teeth, no math problem

tonight.”23 There are now more than 20,000 subscribers to Overdeck’s e-mail list for nightly

problems. Although there have not been any studies confirming the benefits of this routine, the

anecdotal evidence indicates the success of the program. Similarly, though not a formal study,

“Snacktime Math” a program implemented at a summer camp in New Jersey reported data that

21 Picker, “Images of Mathematicians,” 66-70.22 Rochman, “Counting Sheep,” 52-5423 Ibid

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over 70% of primarily low-income students attending the camp improved their math skills in just

six weeks when they solved “Bedtime Math” problems on a daily basis.24

Another example, the Math Circle, which has now spread to many cities across the

country, has created enough appeal for students to wake up early on Sunday mornings to do math

rather than sleep in. The program, founded by Ellen and Bob Kaplan, has a simple formula for

piquing student interest: bring together a group of students, introduce some exciting problems,

and step back.25 Using the Moore method of teaching, a constructivist approach in which the

students make all the discoveries after the instructor merely introduces the topic (and gives an

occasional push in the right direction), the students get the full math experience. Twenty years

after being founded, the Math Circle still brings in enough students to fill sessions four days a

week every semester.26

The success of these two programs demonstrates the ability to stimulate student

fascination in a subject that is widely considered boring, difficult, and unpopular. No matter

where the dislike originated, there is the potential to make the subject more interesting and more

understandable to students, or rather people, of all ages. While there are many possible methods

to eliminate this math aversion, addressing the subject divide head-on should prove to be

effective at increasing student’s math abilities. By combining mathematics with literature and

reading, students will not only learn the material, but also gain a better appreciation of the

subject. The elimination of solely relying on rote mechanics in math classes, and increasing

exposure to (relatively) real-world problem-solving explorations will demonstrate the potential

necessity and beauty that the subject of mathematics can have.

24 Rochman, “Counting Sheep,” 52-5425 Kennedy, “Math Circle.” 9-1026 Kennedy, “Math Circle.” 27-28

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With regards to math education, there are essentially three basic stages into which we can

divide the subject: elementary mathematics, basic mathematic topics, and advanced math

concepts. Elementary mathematics, as its name suggests, revolves around introducing young

children to the various areas of mathematics. Aside from counting and arithmetic, most topics at

this stage are open and conceptual. The overlap between math and literature is much larger at

this level, as many children’s books deal with introductory concepts such as counting, size, or

shapes. The next level, basic mathematic topics, include the common areas each of us studies in

our typical K-12 educational process: algebra, geometry, trigonometry, statistics, all the way up

through calculus. This area, which spans most of our educational math encounters, tends to be

incredibly problematic, as many students lose interest due to the inflexibility of the rules and

formulas in each respective area. There is very little overlap between math and English at this

stage, and the gap between the arts and sciences widens immensely. Finally, the last stage,

advanced math concepts, addresses ideas that few people outside of math majors and

professionals see. Topics at this stage include number theory, topology, numerical analysis,

differential equations, non-euclidean geometry, and more. Once again this stage allows for a

better overlap with the rising popularity of “math melodramas” novels, though there still remains

a lot of room for improved integration into classrooms. While these topics are not typically a part

of K-12 education, there is no specific time or age at which these subjects are (or can be) taught

to students. This means that although it may not be standard practice, it would be perfectly valid

(and even possibly beneficial) for students without a working knowledge of calculus to begin

learning about some of the topics covered in this section.

In addition to the three levels of mathematics education, there are two primary ways in

which a bridge between math and literature can be formed, though one could certainly argue for

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more. The first, and more obvious, connection is through the incorporation of literature already

dealing with math concepts into the classroom. These types of readings can range from

children’s books, to sections of a grad student’s dissertation, all the way up to full-blown

fictional novels about number theory. The second way to combine the two subjects is through a

collaboration between math and English. To do this, ideas and math concepts would be drawn

from literary works and applied to a mathematics classroom. For example, teachers could take an

age-appropriate book, such as The Hunger Games, and create math activities revolving around

the relevant mathematical concepts, like algebra or geometry, based on the text. This example,

along with many others, will be explained in greater detail later. I propose that if all three levels

of math education can improve the relationship between mathematics and literature in the

classroom, students should gain a stronger, more sincere interest in math, helping even the most

math-averted people to acquire an appreciation of the subject.

~~~~~~

Before delving into the possible benefits for literature-infused mathematics, it would first

be beneficial to explore the ways in which reading and mathematics are already intertwined, and

the consequences of this relationship. When we perform or research mathematics, as with any

subject, we require the ability to recognize words and symbols and assign an appropriate

meaning to them; a skill better known as reading. The ability to read is, in a sense, a prerequisite

to perform mathematics. While it is possible to count and learn some basic arithmetic, short of a

genius with unparalleled mental math capabilities, it becomes necessary to write steps down to

present and solve problems which, even at the most basic level, requires reading. Similarly, most

of mathematics is learned, or at the very least printed, in textbook form; a medium through which

a lot of learning can take place. Even if a student rarely references it, their teacher most likely

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relies on some form of hard copy to monitor class progress and ensure appropriate and extensive

coverage of the material. Having a written (and published) reference, such as a textbook, can

help teachers with course design. For example, when learning algebra, an instructor would not

start teaching exponential equations until all necessary pre-requisite topics, such as basic y=x

equations, linear functions, polynomials, etcetera, have all been covered.

What we can take from this relationship is that whether we would instinctively notice it

or not, learning math necessitates the ability to read, making it appropriate to understand the

mental processes involved with reading comprehension. When we read a text, our brains create a

mental representation of the information.27 The most basic, trivial model of this mental

representation is a network of associations, like a tree diagram, with connections between all

related ideas and concepts, the width of the connecting lines representing the strength of the

association. The stronger, and more widely accepted model divides the associations into three

different levels: the surface component, the text-base, and the situation model.28 The surface

component, just as it sounds, is composed of the words and phrases which are encoded in the

brain, but free from their actual meaning. This specification means a representation with a strong

surface component may include exact wording or phrasing, but without any sort of understanding

of the text. The next level, the textbase, contains the meaning of the text as understood by the

reader. This distinction means that the information we take away from the text, whether it be

accurate or full of reading errors, is included at this level. Finally, the situation model is made up

of all the appropriate prior knowledge that helps to connect ideas in the mental representation.

The situation model essentially integrates a reader’s relevant knowledge with the information

that becomes stored in the textbase.29

27 Österholm, Magnus. 2006. “Characterizing Reading Comprehension of Mathematical Texts.” Educational Studies in Mathematics 63 (3) (Nov.): 325-46.

28 Ibid29 Ibid

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This model of reading comprehension, which has been verified by many studies, should

hold true for reading mathematical texts, as well as literature. In fact, a study has shown that the

content of the material makes less difference on reading comprehension than the use of symbols

in a text.30 This difference will be discussed in further detail later on. However, since we can

confidently assume that the reading comprehension of mathematical texts can be similarly

associated with literary texts, it would logically follow that connections can be made not only

with mathematical ideas, but literary ones as well. For example, should a student encounter a

problem in math class similar to one they have come across while reading, when they have

developed appropriate comprehension skills, strong connections and associations should already

exist, helping him or her overcome the distraction of math anxiety and better problem-solve how

to come up with a solution. The benefits of mixing these two disciplines are strong enough that

the National Council of Teachers of Mathematics (NCTM), and the International Reading

Association (IRA), as well as many state standards, often encourage, or even require, reading

across the curriculum.31 They even place a special emphasis on the use and understanding of

specific mathematical language, which can be found in a multitude of age-appropriate books.

When students are exposed to these mathematical terms outside of the classroom, and see them

being used in the real world, it becomes easier to see the applications of the material; an

important step in spurring student excitement about mathematics.

Aside from being a necessary skill to actually perform mathematics, reading can similarly

function in many other ways that can boost student learning, especially in an “inquiry-based”

classroom environment. As more and more teachers begin to utilize constructivist teaching

methods, the added skill of reading can greatly enhance a student’s experience with the material. 30 Österholm, “Reading Comprehension,” 325-46.31 Wallace, Faith H., Mary Anna Evans, and Megan Stein. 2011. “Geometry Sleuthing in Literature.” Mathematics Teaching

in the Middle School 17 (3) (October): 154-9.

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Constructivism is a learning theory that believes humans learn primarily through exploration,

experience, and reflection. Research performed at Cornell University’s Department of Human

Development agrees with this belief; their studies show that most people begin learning in this

manner as young as infancy.32 Cornell’s Tamar Kushnir says that babies formulate questions and

theories, then test these theories and draw conclusions from their findings. They learn by

exploring the world around them through experience. These results align perfectly with the

constructivist theory. Since humans already appear to be learning in a constructivist manner from

birth, it seems appropriate to continue this manner of learning throughout a student’s time in

school, and the math classroom should be no different.

Though it may be difficult to continuously implement, given the amount of information

necessary to learn in courses such as algebra and calculus, when lessons are planned in a

constructivist manner, the type of learning becomes deeper and more enrooted, giving the

students a better understanding than they would receive through mere repetition. Inquiry-based

learning resonates perfectly with the rise in popularity of this learning theory. Inquiry is defined

by Charles Saunders Peirce and John Dewey, two early 20th century mathematicians who helped

reform math education, as “the process of settling doubt and fixing belief within a community.”33

Many teachers are beginning to utilize this philosophy in their classrooms, replacing the

commonly used “techniques curriculum,” which portrays math as a collection of facts and

procedures. This style of teaching reinforces the commonly held myths about learning math

which are counterproductive for learners who see the subject as boring, repetitive, and concrete

in nature.

32 Kushnir, Tamar. “Learning About How Young Children Learn.” Cornell.edu (2011) Ithaca, New York: Cornell University. Accessed October 2013.33 Siegel, Marjorie, Raffaella Borasi, and Judith Fonzi. 1998. “Supporting Students' Mathematical Inquiries Through

Reading.” Journal for Research in Mathematics Education 29 (4) (Jul.): 378-413.

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Inquiry-based learning, on the other hand, encourages students to get involved in the

“experience” of math. As Marjorie Siegel of Columbia University puts it, inquiry learning allows

the learners to “experience and appreciate first hand the ambiguity, nonlinearity, and ‘conscious

guessing’ associated with the mathematical thinking of professional mathematicians.”34 When

teachers take advantage of this capability, they can present math in its natural and true form, one

which involves creativity and problem solving, in addition to the equations and formulas which

are also associated. As demonstrated by the popularity and success of programs such as the Math

Circle, which take full advantage of constructivism, it seems appropriate that math classrooms

that operate in a similar manner would meet equal amounts of success.

A distinct advantage of these classes is that reading opens up a whole world of

opportunities for learning. Language as a whole becomes incredibly important to the learning

process, as meanings and representations are created in the learners’ world. As opposed to

traditional math classrooms where techniques and formulas are merely explained and repeated,

students need to communicate to formulate their own meaning and understanding. Language

becomes more than just a channel through which previously existing knowledge can be

transferred, language becomes a powerful tool.35 As alluded to earlier, in a traditional classroom,

reading is often viewed as an obstacle; though it is necessary to reach the “expert’s message,”

one can only interpret this message if they have proper reading skills. Writing, then, is the means

of demonstrating what has been learned. On the other hand, in the appropriate classroom

environment, specifically a more inquiry-based one, reading, writing, and even speaking can take

on new roles which will actually enhance a student’s experience with the subject, giving them

even more knowledge than other classroom formats could offer.

34 Siegel, “Students' Inquiries,” 378-413.35 Ibid

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So what is “knowledge” exactly? Looking to the study of the natural sciences, and the

process involved, most modern scholars have rejected the belief that knowledge is a stable mass

of information, and instead replaced it with the belief that knowledge is a “dynamic process of

inquiry in which the doubt arising from an anomaly sets in motion the struggle to settle doubt

and fix belief.”36 The scientific processes of learning and forming knowledge can apply just as

well to mathematics as any of the other sciences. So classes which operate through inquiry allow

the students to be active members helping discover knowledge in the field of mathematics. Much

like labs in science which involve testing hypotheses and experimentation to discover new

knowledge, so too can a math class allow students to discover knowledge for themselves. It only

requires an environment that encourages this type of learning. The assumptions which define a

classroom as inquiry-based are:

1- Knowledge is reflexively constructed through a process of inquiry that is

motivated by ambiguity, anomalies, and contradictions and undertaken within a

community of practice

2- Learning is a generative process of meaning-making, requiring both social

interaction and personal construction in a purposeful situation.

3- Teaching is establishing a rich environment for inquiry and establishing the

conditions that support a community of learners.37

Of course, these assumptions must be understood as contributing to a long-term

engagement with the subject. While brief encounters will still be beneficial, it is through the

continued implementation of this process which will transform students from passive learners

into active participants in unveiling mathematical knowledge. Similarly, it should be noted that

executing this philosophy requires a lot of effort on the part of the teachers who have to carefully 36 Siegel, “Students' Inquiries,” 378-413.37 Ibid

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plan lessons, while also being very flexible and patient as the students make most of the progress

on their own. And, because the popularity of this belief is relatively new, there are not as many

resources to help teachers, as there are for other learning styles. However, the benefits from this

type of learning still remain, and students will have a much better appreciation and

understanding of the subject upon completion of the class.

Dr. Siegel proposes inquiry cycles as one possible way to help cultivate this type of

learning environment. An inquiry cycle is comprised of four stages: problem sensing, problem

formation, search, and resolution.38 As mentioned above, doubt plays a major role in this process

as anomalies, and contradictions lead to questions and eventual exploration of the topic. In a

classroom, the students become the focal point. They are the primary members and explorers

who all share responsibility in helping decide how to proceed with the inquiry, and reaching

eventual conclusions from their exploration. Expanding on the basic stages of an inquiry-cycle to

be more accommodating for mathematics, Dr. Siegel presented the steps of a “mathematics

inquiry cycle” to be used in a classroom: “setting the stage; developing and focusing one’s

question; identifying appropriate approaches, resources, and tools for exploring the question;

carrying out the research; collaborating with other inquirers; reflecting on and expanding the

results of one’s inquiry; communicating with outside audiences; identifying problems and

planning strategy instruction; and offering invitations for new beginnings.”39 As the study was

carried through, these steps were then regrouped and morphed into four chronological phases,

“Setting the stage and focusing the inquiry, carrying out the inquiry, synthesizing and

communicating results from the inquiry, and taking stock and looking ahead.”40

38 Siegel, “Students' Inquiries,” 378-413.39 Ibid40 Siegel, “Students' Inquiries,” 378-413.

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So how does reading come into play in these types of math classrooms? What does

reading have to do with inquiry-cycles? In the study performed by Dr. Siegel, along with two

University of Rochester Professors Dr. Raffaella Borasi and Dr. Judith Fonzi, they operated

under the assumption that literacy skills of reading, writing, and talking offer a range of

opportunities for students to become engaged in the inquiry-cycle. In a 1975 study, linguist Dr.

Michael Halliday found that language serves at least seven different functions in our lives

(instrumental, regulatory, interactional, personal, heuristic, imaginative, and informative).41 Our

education system, however, tends to heavily emphasize the informative function, allowing the

remainder of the functions to fall on the wayside. Language educators, as a result, have begun to

call for instructional environments that provide students with more opportunities which allow

them to use reading, writing, and talking for purposes that reflected the nature of language

outside of the school setting.

In a similar study, linguist Dr. Shirley Heath identified a variety of functions that reading

and writing serve outside of classroom settings such as building and maintaining relationships,

learning about the news, enjoyment, or accomplishing an array of simple tasks (paperwork e.g.).42

Most of these functions are taken for granted in our daily lives. In the classroom, however, the

roles of reading and writing tend to be aimed primarily at accomplishing the same repeated tasks,

namely we read for meaning, and write to communicate our learning. Again, as a result,

language educators called for learning environments which helped bridge the gap between

reading and writing functions in the outside world, and in a classroom setting. Bridging both

gaps addressed in these studies demonstrate that the uses of language and literacy in math

classrooms is far more expansive than was ever previously considered. Reading, of course, as

argued in this essay, needs to be expanded beyond the typical notion of reading that is applied in 41 Ibid42 Ibid

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math classrooms. It goes beyond learning mathematical symbols and gaining strategies for

tackling word problems, to encompass all sorts of math-related texts including but not limited to

historical essays, diagrams or even literature.

Looking at one case study of an inquiry-cycle used in a classroom, the added value from

reading becomes incredibly apparent. In fact, after the completion of the study, there were 30

various functions of reading that were identified, 27 of which were all present in just one of the

three observed courses: the narrative of the “Taxi-Geometry.”43 This unit was a part of a

semester-long course entitled “Alternative Geometries” offered at an alternative urban public

high school. The students were 10th – 12th graders who had completed at least two high school

level math courses, and had all been previously exposed to reading strategies encouraging sense-

making and discussions. Taxi-geometry, as suggested by its name, is

made up of a grid where only horizontal and vertical movements are

allowed (like a taxi-driver navigating blocks in a big city).44 Although

this world seems trivial enough, as it essentially simplifies the real

world, many aspects of geometry that we take for granted no longer

hold. For example, the shortest distance between two points is rarely a

straight line (see figure 1). The shortest path would only be a straight

line if the given points were perfectly vertical or horizontal to one another, otherwise, alternative

steps up or down would be necessary, and often times there would be multiple “shortest paths.”

Because this situation is easily graspable, the student’s challenges arise from their mathematical

understanding of definitions, formulas, proofs, and truth, rather than from a conceptual

understanding of the material. The simplicity of the taxi-geometry scenario similarly allowed for

more time on reflection, and was later used as a springboard for other mathematical explorations 43 Siegel, “Students' Inquiries,” 378-413.44 Ibid

Figure 1: As can be noted, all three paths above are of equal length, thus all three could be considered the shortest distance

18

later in the semester. Of course, it should be noted, that this structure was designed for these

exact purposes.

In the first phase of this process, setting the stage and focusing the inquiry, students were

asked to answer questions which made them reflect on some mathematical concepts and issues

that they most likely wouldn’t encounter on their own. Already, a function of reading

(challenging student’s initial concepts and knowledge of the topic being explored) reared its

head.45 The next step in this phase involved reading even more directly. The class spent several

periods reading an essay, “Beyond Straight Lines,” by J Sheedy, which discussed his own

explorations with the subject. In the essay, Sheedy even addressed his discomfort with the idea

of alternate geometries and gave a reassurance that this discomfort is a natural stage in the math

exploration process. This reading not only demonstrated to students that it is normal to encounter

hesitations and concerns in the process they are about to engage in, but it also introduced the

subject to the class – generating interest and knowledge of the subject they were about to

explore.46 Both of these demonstrations were later categorized into 2 of the 30 formal reading

functions in the inquiry-based learning math classrooms.

Rather than reading the essay at once in its entirety, the students read smaller sections at a

time, using specifically-chosen assigned reading strategies, such as writing a journal response, or

reading aloud to another student, to help them benefit the most from the experience. Some of the

following notes from students after the completion of the exercise demonstrate the “richness of

the thinking generated by this reading activity.”

Jolea – Who is to determine the accuracy and what becomes law in math?

Math is humane just like us in the respect that it changes because it is not always

complete and accurate45 Siegel, “Students' Inquiries,” 378-413.46 Ibid

19

Char – “Completeness and perfection are ideals” [-] that kind o struck me as really

Interesting. It’s true now that I think of it, but I never realized it before.47

The next reading in the class was a fictional story, “Moving Around the City,” again by J

Sheedy, where the protagonist navigates a grid-patterned city and encounters several problems,

specifically the non-intuitive consequences of the geometry in this world.48 Rather than have the

students merely read the whole story again, the solutions to the problems were removed from the

text, allowing the students to try and solve them for themselves first, and again write a journal to

reflect on their experience. During the discussion of reflection, one student claimed that this

geometry was the same, just with a new rule – leading to the first in depth exploration of the

subject – is this geometry our normal geometry with a new rule, or a completely new one by

itself? The proceeding discussion led the students to their first inquiry, “what do familiar shapes

look like in this world?” Through reading and responding to the appropriate texts, the students

stumbled upon a subject that they found exciting and worthy of further investigation,

demonstrating the ability of literature to grab students’ interests and, again, revealing another

formal function of reading in a mathematics classroom; to generate specific questions and

conjectures, and find resources to help make sense of these conjectures.49

Soon the class came to a fork in the road, and stumbled upon an incredibly important, and

oftentimes necessary, skill to have in mathematics: interpreting a definition. The class, trying to

make sense of a circle in this world, encountered the textbook definition that defined a circle as

“a set of points in a plane that are a given distance from a given point [the center] in a plane.”50

Of course, this definition also needed to take into account the means of measuring distance in

this world, which again meant travelling only in straight horizontal or vertical lines, and not “as

47 Siegel, “Students' Inquiries,” 378-413.48 Ibid49 Ibid50 Siegel, “Students' Inquiries,” 378-413.

20

the crow flies.” Eventually the students reached a consensus about the new image of a circle in

this world (see figure 2). This investigation forced the students to think deeply about the use of

definitions in mathematics, and as mentioned by Sheedy, doubtful results; i.e. for example,

following the definition of a circle, the resulting shape is no longer round.51

Student’s eventual excitement with this discovery led them to the formulation of more

inquiries with regards to what other shapes would look like in this world, and what other means

of measuring distance exist. Little did they know, this seemingly simple concept of measuring

distance is actually a branch of topology where different systems of measuring distance are

referred to as metric spaces. Keeping on top of student interest, the instructor provided more

readings about these subjects, several of which left much to be desired by the students who then

took researching into their own hands.

Finally, holding true to a constructivist-teaching format, the students were given control

over the conditions of their final project for the unit. One student built a geoboard, two others

constructed a “taxi-globe” by rotating a taxi-circle around the vertical axis to construct a three

dimensional figure. Another student, inspired by the story “Moving Around the City” wrote his

own story about a similar city and read it aloud to his classmates. In all of these examples,

students used some form of “reading” (being loose with the definition of reading to also include

diagrams and other nonverbal texts) to come full circle and demonstrate their understanding of

the material and present results from their investigation.52

Students were then presented with a few more articles to read in tandem with their

reflection on the experience. They were asked to answer questions about the readings, and the

impact they had on their exploration. Reading strategies that the teacher had encouraged were

brought up again as a reminder, and then they were asked how effective they felt each strategy 51 Ibid52 Siegel, “Students' Inquiries,” 378-413.

21

had been. The answers to the question “What did reading this story [Moving around the City] do

for us?”, which were posted on a wall along with much of the student’s other work, included

responses such as: helping understand the geometry better, made the student think about how a

city is planned, and helped pique student interest by putting them in the shoes of the protagonist

trying to solve the encountered problems.53 During this discussion, the question of what would

happen if the surface were a sphere, as opposed to a flat grid led perfectly into another inquiry

cycle – one in which students even drew from their own sources to help make sense of the new

problems posed in the readings handed out by the instructor. Thus the new inquiry cycle began.

Along the entire course of the taxi-geometry inquiry cycle, many more functions of

reading were discovered and recorded, only a few of which were alluded to in this overview of

the study. In fact, all but 3 of the 30 defined reading functions were identified somewhere in the

three different inquiry cycles observed for the study, and both of the other two other cycles (not

discussed in this paper) included at least 15 functions.54 The abundance of these functions

indicates the important connection between reading and mathematics that is present, especially in

inquiry-based classrooms.

Upon a completion of the study, the list of observed functions was grouped into two

distinct parts: chronological and embedded. The chronological functions were connected to

specific stages in the cycle, whereas embedded functions cut across the stages and were present

throughout.55 Both groups could find a place in any mathematics classroom. One could certainly

argue that many of the functions primarily help to construct and carry out the inquiry cycle, for

example using articles to inspire students to create their own investigations, or reflecting on the

inquiry process. While some of these may not be as useful outside the setting of an inquiry cycle,

53 Ibid54 Siegel, “Students' Inquiries,” 378-413.55 Ibid

22

there were many other functions which transcend the classroom context. Many would work just

as well in a traditional, non-constructivist classroom. Reading to both generate interest and gain

background knowledge, for example, was prevalent in the study and could easily be generalized

to a larger audience when trying to introduce new ideas or concepts to students. Similarly, many

of the readings helped encourage students when they ran into doubt or frustrations, since they

were shown many other people who encountered similar difficulties. And in case those two

benefits weren’t enough, the readings also addressed a deeply-rooted problem with mathematics

by encouraging students to “rethink their conceptions of mathematics and learning mathematics

by appreciating the humanistic dimensions of this discipline.”56 In spite of the cultural

perceptions of the subject, through reading, students can begin to see math in a new light; as a

discipline which goes beyond mere formulas and calculations.

After completion of the study, during a reflection on their work, the researchers Dr.

Siegel, Dr. Borasi and Dr. Fonzi also found that the functions of reading used in the math

classroom also aligned with many reading theories as well. For example, Dr. Louise Rosenblatt’s

transactional theory of reading, a reader-response theory which places importance on individual

reading and interpretation, “provide[s] an apt description of reading experiences identified in

Setting the Stage and Focusing the Inquiry.”57 At this early stage of the Inquiry-cycle, the

primary objective of the texts was personal exploration through prior knowledge and personal

experience which, ideally, leads to questions for inquiry. These types of reader-response theories

are also present in the final stages of the cycle dealing with reflection and potential future topics

for investigation. Since everyone interacts with the texts in a unique manner, each individual’s

reflection and interests will lead to a diverse exploration of the topics being addressed. And

56 Ibid 40057 Siegel, “Students' Inquiries,” 378-413.

23

similarly, each one of the embedded functions was associated with specific reading practices laid

out in earlier research.

The problem with this observation, however, lies in the fact that our education system

places the most (if not all) of the emphasis of math education on what is essentially the carrying

out the inquiry stage. The reading which takes place during this time tends to be more technical

and text-based, which is where most analysis of math-reading takes place; researchers spend

most of their time trying to figure out how students understand the content of their math books

and similar texts. While this type of reading is important, and does account for most of the

reading which takes place in traditional math courses, it doesn’t account for many of the possible

benefits which can be gained if we implement these reading functions in the classroom. There is

still much to investigate with regards to bringing literature into the classroom, but already we

have shown that there are many more connections between the two subjects than most people

would imagine.

That being said, there still remains the concern that reading only works in an inquiry-

based setting. Before directly addressing that issue, it should be noted that in this example,

reading was not a supplementary activity but rather the primary focus. It was the entire means of

carrying out the inquiry cycle, thus the classroom learning only occurred because of the reading,

and without it, no investigations would have taken place. Reading played a role at each stage in

the cycle, helping drive the investigation forward and continually keeping the students engaged

with the material. So when the students in this course gained an appreciation for the subject and

became more fluent in mathematics, they did so solely through the context of reading. One could

certainly argue that this context is specialized, and one could similarly argue that reading isn’t

necessary for students to learn. Both of these arguments are valid, though it is clear that in this

24

case study, the reading was more than effective, it was the entire foundation of the learning, and

to ignore the numerous benefits to be gained would be foolish. Teachers could certainly continue

teaching without implementing reading into their classroom, but I believe that by doing so, they

are missing out on major instructional opportunities for their students.

~~~

Having explored the pre-existing connections between reading and mathematics, I would

now like to begin a more thorough investigation into the possible benefits of this relationship. To

recount, I want to examine mathematics classrooms at the three primary stages in which I

divided our mathematical learning into: elementary, where we are just beginning to grasp

concepts such as shapes and numbers; basic math, which includes standard topics such as algebra

through calculus; and advanced math which encompasses topics like number theory or

combinatorics. Similarly, I argued that there are a couple ways, at each level, in which literature

can be brought into the classroom, and I would now like to begin exploring this claim.

Perhaps the most important time period to begin developing student’s fascination with

mathematics would be at the elementary level. If students are introduced to mathematics in ways

that draws them in, it will be much easier to keep them interested. This would ease the challenge

of convincing students that math can be interesting after they have developed a dislike for it. In

fact, making this introduction at the elementary level actually allows for some of the most

interesting elements of math to be explored in fun and non-formulaic ways. Early on in students’

mathematical experience, each aspect they encounter is new and exciting, and their

understanding of arithmetic will be minimal at best. This blank slate of knowledge offers up the

perfect opportunity to make math appealing, giving students a positive first experience.

25

The typical pattern of learning mathematics, specifically arithmetic, begins by using

physical objects, allowing the students to interact with the world while they are learning.58 When

children can feel, see, and interact with the concrete objects, they can more easily make

connections with the ideas. Early on in their education, most students will have participated in

some sort of activity which involved counting and moving around blocks or tiles, helping them

understand the connection between the abstract numbers and the physical objects they are

manipulating. The idea of “two,” for example, only makes sense if kids have two objects to

associate the idea with (aka two things to count). The next common step moves away from

concrete objects, onto pictorial representations.59 This small step is the first level of abstraction

from the concrete world. The students have images they can count and manipulate, but the

images are not actual things, a difference which takes time getting used to.

Continuing along this process of learning mathematics, children are introduced to another

abstract concept, numerals (1,2,3, etc).60 Most often, the first interaction with these symbols deals

with counting purposes – assigning meaning to the arbitrary figures – because the leap from

physical/countable objects to a symbolic representation can be a big one. In fact, even after many

students have become familiar with this abstraction, many of them will continue to rely on

concrete visualizations as they make the move towards basic arithmetic. Greg Tang observed

students learning addition in classrooms with dominoes were actually counting the number of

dots on the dominoes to reach the final sum, rather than simply adding the numbers together.61

Likewise, we can all attest to witnessing young children still counting on their fingers rather than

performing mental math in their head. This step is, of course, the final prerequisite to mastering

basic arithmetic, and it requires a wholly abstract grasp on what is occurring. There is no

58 Lowe, Joy L. Matthew,Kathryn I. 2000. “Exploring Math with Literature.” Book Links 9 (5) (05): 58-59.59 Ibid60 Lowe, “Exploring Math,” 58-59.61 Tang, “WORRY Out of MATH,” 44-45.

26

physical connection between the symbol 4, and four apples on a table, aside from the meaning

which human kind has given to the symbol 4, or the word “four.” This should be obvious since

the words “cuatro,” in Spanish, or “cat” in French both have the same meaning, but share no

physical or harmonic similarities.62 Or the various ways we can print the number 4, such as

roman numerals IV, or the Chinese character 四, the only similarity between these symbols and

sounds is the significance of their human-assigned meaning. The process of performing

mathematics has made its first step entirely out of the concrete world, and into one of abstraction.

The importance of this fact is that very early in the learning process, visuals and imagery

already play major roles. This means something as simple as using pictures, which appeal to the

students, can help draw them in and keep them interested.63 And what better way to introduce

these math concepts visually than in children’s literature? Not only can these books use

illustrations to grab student’s attention, they can also place the math concepts in real-world

situations that the characters find themselves in. As mentioned above, this method of combining

math and English involves literature based on mathematical concepts, and there is far from a

shortage of these types of books. Another major benefit is that there is sure to be some book on

every introductory math concept; from counting to pattern recognition, from shapes to grouping

digits, some children’s book covers it. In fact, there is a book series dedicated to just this goal of

blending math with literature. The Hello Reader! Math series contains dozens of books for

various levels of math ability, covering preschool through first grade.64 And each children’s book,

regardless of whether they are in the Hello Reader or not, comes fully equipped with illustrations

to further demonstrate the math concepts being introduced.

62 Saussure, Ferdinand de. Course in general linguistics. New York: Philosophical Library, 1959: 80-90

63 Tang, “WORRY Out of MATH,” 44-45.64 Hopkins, Gary. "Math and Reading Do Mix!" Education World.

27

A perfect example of this type of synthesis for elementary level mathematics is Big

Numbers and Pictures that Show Just How BIG They Are, by Edward Packard. Dealing with the

concept of large numbers, the book follows the common thread of a pea to illustrate how big

numbers can get.65 Pete, the main character, first sees one individual pea on a plate, followed by

10 peas on the next page, then 100, until there are 100,000 peas overflowing onto the table. This

visualization brings the readers back to the stage of pictorial representations, and even though

they are unlikely to count all 100,000 peas, the image still drives home the message; 100,000 is a

very large number. To further illustrate this concept, Packard has Pete, accompanied by his dog

and cat, travel out to the moon (240,000 miles away) and eventually even further. The threesome

goes far enough into space to eventually allude to the idea of infinity. At a certain point, after

travelling far enough into outer space (10 to the 27th power miles away) they all look back

towards the Earth, which now looks like the size of a pea.

While travelling in a space ship thousands of miles may not be something children are

going to experience in their own daily lives, many other books contain elements which show the

kids how people encounter math in everyday life as well. Seeing the math concepts in the real

world helps them understand math and its importance. It is worth noting here that it is essential

for the books to first be read for pleasure, and to later introduce the mathematics and problem

solving.66 This way the books can be enjoyed for what they are, fun children’s books, rather than

becoming a chore. If, however, children aren’t interested in math-related books at a young age,

there is still another strategy to introduce the same ideas; taking non-mathematical books and

extracting mathematical concepts from them.

With a little bit of creativity, any children’s book can become a source of inspiration for a

number of math related activities. For example, a favorite among many children, The Giving 65 Ibid66 Lowe, “Exploring Math,” 58-59.

28

Tree, by Shel Silverstein has a universal appeal across the world, and has been translated into

over a dozen different languages. Using the appeal of this book as a springboard, teachers and

parents alike could create activities stemming from events in the story. For instance, at one point

the young boy in the story picks all of the tree’s apples to sell them for money. After reading the

book, an activity dealing with apple counting or figuring out the finances of selling the apples

could easily rear its head. The benefits of this synthesis is that no matter what types of books

interest a child, some element of math can be found inside it. Utilizing the internet can expedite

the whole process, as thousands of sample activities already exist, and can be found merely by

Googling the book’s title and “math activity” afterwards. Hundreds of ideas and samples can be

found with almost no effort at all.

As mentioned earlier, this stage of learning is the most crucial, but also the easiest to

improve. Greg Tang acknowledges that when he visits schools and takes polls of who loves

math, nearly all young kids will raise their hands. It is only when they get older and begin the

second stage of math understanding that they begin to dislike it.67 What this indicates is that at

one time, almost everyone loves math. If we take advantage of this by encouraging students to

not just read, but to read math related books, then the beauty and mystery of abstract math

concepts, and the fun of problem solving that arises from these concepts will present themselves

to the children. Or, on the other hand, if teachers can tailor assignments towards students’

previously existing fascinations, they can construct math related activities from non-

mathematical stories, exhibiting some applications in the real world and, again, demonstrating

the fun that exists in the subject.

The importance of enhancing a child’s pre-existing fascination with the world of

mathematics increases due to the following stage of mathematic development, namely the basic

67 Tang, “WORRY Out of MATH,” 44-45.

29

mathematics concepts. This stage includes all the math one is likely to study from roughly 2nd

grade through the end of high school: algebra, geometry, trigonometry, statistics, pre-calculus,

and eventually calculus. Some, or rather many, students will not even complete all of these

foundational courses, and for those who do, it is very often the case that they don’t finish them

all until college. Unfortunately, this stage is where most students begin to despise math, and the

separation between students who “understand math” and those who don’t becomes established

and solidified. With a firm introduction to mathematics through children’s literature, students at

this stage should no longer need to be convinced that they can enjoy the subject, rather they will

merely need to maintain that viewpoint. Again, using literature can be a way to keep students

interested and engaged.

Before addressing the benefits which can be gained at this level of math education, I

think it would be helpful to first look into the reason many students begin to dislike the subject

all together. As mentioned earlier, the cultural view on math remains the same, “math is

difficult.” When adults claim to dislike math, their children are very likely to adopt this similar

attitude, meaning it can be difficult to entice a student who has pre-determined they will not

enjoy the subject.68 Especially after the enchantment of math at younger ages wears off, students

are more prone to join the masses in the revolt against the pleasures of math. Similarly, because

of the nature of most of these concepts, it can be very difficult for a teacher to inspire learning in

the students. There are so many rules and formulas to memorize, all of which lack the attractive

creativity of real mathematics. As Greg Tang, and the Kaplans have hypothesized, it could very

well be the manner of instruction that makes kids lose interest.

Even if the problem lies more with the content than the teaching procedure, there is still a

necessity to revise and improve the method of teaching mathematics, specifically, with regards to

68 Hopkins, Gary. "Math and Reading Do Mix!"

30

reading math textbooks. In a 2006 study, Magnus Österholm of University of Umeå, revealed

that reading mathematical texts with symbols requires a different set of skills than normal

reading comprehension.69 Their investigation was inspired by the universality of textbooks used

to teach mathematics, and the common use of symbols inside these texts. Prior to this study,

most studies of math comprehension focused primarily on the problem-solving aspect of math,

and tackled their research with the mindset that reading more often presented an opportunity for

misinterpretation and misunderstanding. What Österholm wanted to argue, on the other hand, is

that reading comprehension could be viewed as an essential part of math ability, rather than a

weak relation of it.

The procedure of the study was to let students read a one-page math text about group

theory with either symbols or natural language explanations, and were all also given a one page

historical text. After each reading, the subjects were then given a test of their reading

comprehension. The data collected took into account outside factors such as prior knowledge,

college or high school enrollment, etc., and they judged the results by recreating “mental

representations” (as defined earlier) based on student responses to the post-reading questions.

The results showed that reading comprehension of the math text without symbols was highly

correlated with reading comprehension of the history text, but not related to the math text with

symbols.70 What this relationship demonstrates is that the content of the text makes no difference

with regards to comprehension, and suggests that we need another whole skill set when we read

texts with symbols in them (essentially every single math textbook).

Although the reasons for this difference is unclear, there are several possibilities that

Österholm presented. One of them relates to our expectation when we see symbols in a text.

When symbols are on the page, we more often expect some sort of procedural demonstration to 69 Österholm, “Reading Comprehension,” 325-46.70 Österholm, “Reading Comprehension,” 325-46.

31

follow, which we internalize differently. Another hypothesis was since symbols can be used in so

many different contexts, our brains need to figure out which context we find ourselves in at

every encounter. Or another possible explanation is because humans have a tendency to skip

over the parts of texts containing symbols with the intention of returning to them later. These

theories all have some potential truth to them, however as Österholm admits, more research on

this would be necessary before reaching any solid conclusions. The article did make clear the

researchers do not mean to suggest that math texts should only be printed in natural language.

The use of symbols in mathematics, Österholm says, is necessary and a major advantage of the

subject. Rather than changing our texts, it would be more beneficial (and practical) to recognize

the other skills necessary, and help students to develop this ability. In this way, reading and math

would both benefit. I propose that bringing more literature into math classrooms would be an

excellent way to increase this skill.

Following the same pattern set out, the first strategy to synthesize math and literature is to

have students read texts that revolve around and introduce mathematical concepts. The trouble

here is that often books of this type are not nearly as abundant as with introductory level

concepts. Algebra does not always make for the most exciting plot lines, whereas the extensive

amount of freedom in young children’s books make for an easier synthesis. It has been done,

however. The book, “Do the Math; Secrets, Lies and Algebra,” by Wendy Lichtman has taken a

stab at this daunting task.71 The main character in the book, Tess, likes the concreteness of math

but has her world shaken a bit with the introduction of variables in her classes. But, as she begins

to learn algebra, she begins applying things she learned in class to her everyday life. Though the

book is a little juvenile (it would probably not be well received by high school seniors for

71 Pestro, Annie. 2008. Mathematics Teaching in the Middle School 14 (1) (AUGUST): p. 63.

32

example), it does introduce algebra in a purely literary form. Not only does it provide an

opportunity to springboard out of the book into a discussion of algebra, it also demonstrates real

world applications of the subject. This means that before students have time to ask, “when will I

ever need to know this,” they will already have some examples floating around in their head.

At the intermediate stage of math development, the line between books about math

concepts, and extracting math from non-mathematical texts becomes blurred. The concepts being

learned do not necessarily inspire very exciting stories, but minor examples of math can be found

in an array of books for all reading levels. Whether these should be grouped together with the

literature about math, or with examples from non-mathematical texts is not a crucial distinction

though, as the eventual goal remains the same. Many books are full of mathematics, which could

help inspire and interest in mathematics of some sort. Some examples include The DaVinci

Code, Flatland, or even Sherlock Holmes.

To look specifically into one area of mathematics, geometry, there are numerous

examples of age-appropriate books chock full of real world examples of math problems. Take,

for example, Sherlock Holmes, specifically the story Adventure of Musgrave Ritual. In this

particular story, Sherlock Holmes uses geometry to help recreate one of his old classmate’s

family rituals, to better understand its significance. Using mathematical terms such as “parallel,”

“fixed point,” and even “trigonometry,” the story makes no attempt at hiding the importance of

math in solving the case. Actual geometric calculations are even used in the story itself; “If a rod

of six feet threw a shadow of nine, a tree of sixty-four feet would throw one of ninety six, and the

line of the one would of course be the line of the other."72

Through these types of stories, many teachable moments can arise on their own, helping

to spark students’ interests. Better yet, when the math concepts can already be found in a good

72 Wallace, “Geometry Sleuthing,” 154-9.

33

story, the material can be seamlessly woven into the classroom. These stories show how math

terms can be used naturally outside of the textbook setting, and the multitude of genres allow all

students to find something that resonates with their pre-existing interests. One way to utilize

these texts for their mathematical connections is through reading strategies, such as “coding the

text.” This reading strategy directs students to make notes, predictions, and other connections

while they read, often encouraging them to use their own math knowledge to problem-solve

before the teacher even becomes involved.73 In the series Crime Files: Four-Minute Forensic

Mysteries, when students used text-coding to record important events and approaches to solve

the mysteries, the majority of the noteworthy clues were math-related – meaning that the

students are already beginning to recognize math concepts and their importance in the real world.

Going beyond the in-text applications, however, many other works can inspire other activities to

welcome an even stronger understanding and appreciation of the field of mathematics. In one

example, inspired by the mystery novel Artifacts, teachers Dr. Faith Wallace, Mary Evans, and

Megan Stein set up Cartesian coordinates inside their classrooms and divided the students into

small groups to explore the importance of this seemingly simple concept. They then asked the

students to complete basic tasks such as using coordinates to measure distances (without a ruler),

comparing their results based on the different measurement sizes, and determine the significance

of defining the location of the origin on a coordinate system.74 Activities like this not only allow

the students to recreate the mathematics they encountered in the stories, but also get them

involved in the action of real-life problem solving and applications of mathematics.

Even in middle school or high school when English and mathematics are separated into

completely different classes, teachers still have the opportunity to co-plan lessons and take an

interdisciplinary approach; something that is already highly encouraged in Middle Schools 73 Ibid74 Wallace, “Geometry Sleuthing,” 154-9.

34

across the country. As the National Council of Teachers of Mathematics (NCTM) says,

opportunities for interesting math problems can be found in all sorts of everyday experiences,

including reading.75 This inspired Donna Christy, EdD from Boston University, and her

colleagues at Rhode Island College to compile a list of books, and math activities stemming from

them, with hopes of inspiring other teachers to follow suit. The way they see it, both subjects will

benefit from this relationship. The activities and assignments would enhance the experience of

the students who have read the books, and potentially motivate those who haven’t to read it on

their own.76 Since these activities do not require the students to have already read the books – as

long as the teachers are careful not to spoil anything from the story – both English and math

teachers alike have something to gain. As Christy puts it, integrating math and literature presents

the opportunity to “ignite the imagination and creativity of students and teachers.”

The four examples that are given in their article are The Westing Game, The BFG, The

Red Pyramid, and The Hunger Games; none of which make explicit references to mathematics,

unlike the texts mentioned above. The sample activities, though, still meet both NCTM standards

as well as Common Core State Standards (listed in the article), demonstrating the possibility of

beneficial activities inspired from the text.77 Looking at the Hunger Games, for example, Christy

and her colleagues used a scene where two characters, Katniss and Rue, are hiding in the treetops

with their gear and food. The activity sheet provided, which prints the passage on the top, then

creates a scenario where their total weight is being balanced between two halves of the tree. The

student’s job is to figure out the unknown weight of the food and supplies when the weights of

Katniss, Rue, water, medicine, and weapons are given. This basic algebra problem hardly

resembles the formulaic equation sheets that most students would likely be used to seeing,

75 Christy, Donna, Christine Payson, and Patricia Carnevale. 2013. “The Bridge to Mathematics and Literature.” Mathematics Teaching in the Middle School 18 (9) (May): 572-7.76 Christy, “Bridge to Mathematics and Literature,” 572-7.77 Ibid

35

making the assignment more enjoyable and less of an abstraction. As a more concrete example, it

will help students better understand, since, as discussed previously, math is first learned through

real-world examples before it becomes abstracted. Returning occasionally to tangible problems

will help reinforce the concepts being taught.

As previously stated, this stage of math development is the most important for students

learning the subject. The knowledge that the students acquire in algebra, geometry, and even

calculus will be utilized in all sorts of areas that many would never have even considered. Nearly

all sciences, computer programming, finances, and even basic business functions require some

level of mathematical understanding; making sure that students don’t take for granted the

information they learn during this stage is crucial to their futures in mathematics. The NCTM

encourages reading across the curriculum, and the reason for this is most likely because reading,

when used appropriately can inspire and enhance student’s experiences in almost all disciplines.78

Programs such as the Math Circle have proven the possibility to stimulate student interest in a

subject that many of them will grow to despise. Since attendance at a Math Circle program is not

an option for every student, as the cost of attendance is expensive and the program locations are

limited, teachers need to utilize other possible means of getting students’ fascinated. Co-

curricular activities, inspired by reading, are this great alternate option that can be utilized by any

teacher in any location.

If student’s are lucky enough to move beyond these foundational courses, or able to take

more advanced/pure math courses simultaneously, then the use of literature becomes less of a

chore, and instead helps enhance the learning by addressing some, for the most part, already

thought-provoking ideas. As David Foster Wallace mentioned in his literature review, the

foundational courses are boring and formulaic. While reading literature may help make the

78 Wallace, “Geometry Sleuthing,” 154-9.

36

material more tolerable (for those who don’t enjoy it already), the more advanced stages of

mathematics don’t often require outside sources to make the subject interesting; the material is

already interesting. The skills used at this level include problem solving, imagination and

creativity, as opposed to memorization and calculation. For many people, these topics may no

longer even resemble math. The perception of mathematicians is so misconstrued because people

rarely encounter this aspect of the subject due to the inconvenience of wading through all of the

foundational learning in the intermediate level. What this means for advanced math topics, then,

is that the use of reading can only help to inspire new ideas, enhance explorations, and address

incredibly fascinating ideas already present in the respective fields. Thus I would argue that at

this level of mathematics, though rarely required in an upper division course, classes could

certainly still benefit by integrating math-related literature into the classroom.

For some math-averted students, there already exists the perfect amalgamation of math

and literature at Arcadia University in a course that can either fulfill a core requirement or be

taken out of pure enjoyment. The class “Truth and Beauty: Mathematics in Literature,” which

counts as either math or literature credit, was initiated by Marion Cohen who claims that while

science has led to the genre science-fiction, math has similarly led to an analogous genre.79 The

goal of her course is not to teach mathematics through literature, but rather to use literature to

cultivate an appreciation for the subject, with hopes that students will learn a little bit of math

along the way. Going beyond novels and stories, Cohen even utilizes poetry in the second half of

the semester. She claims, surprisingly, that the amount of material to draw from, for both fiction

and poetry, is incredibly extensive. She has taken things from various anthologies of the genre

such as Fantasia Mathematica, and Strange Attractors: Poems of Love and Mathematics.80 The

79 Cohen, Marion D. 2013. “Truth and Beauty: Mathematics in Literature.” The Mathematics Teacher 106 (7) (March): 534-9.

80 Cohen, “Truth and Beauty,” 534-9.

37

fact that so many examples exist allows her to pick from an array of topics, as well as select

better-written pieces of literature so that neither math nor English has to suffer as a result of the

relationship. Similarly, since the course has no math prerequisites, and many students often have

an aversion to math, Cohen tries to pick math topics which are not too complicated or obscure,

but at the same time are not so trivial that the math majors enrolled in the course become bored.

Each unit begins with reading either a piece of fiction or a poem, which Cohen prints for

every student, so no textbook is required. Initially, the emphasis is focused on the literature

aspect of the story, as that is more universal and easier to discuss. Then, after the first read-

through, the actual math ideas make their way to the foreground. This occurs first in homework

assignments and eventually in class discussions. Each piece of literature that students read comes

with two assignments, one literature-based one math-based, and the class culminates with a

project where students compose their own poem or short story dealing with math. In her four

years teaching the course, Cohen has found that the whole classroom atmosphere is brightened

by the presence of literature. Her lectures and the class discussions have become so fruitful that

she believes that this strategy (using literature to stimulate student math-interest) could be used

in any math course at any level. Cohen essentially argues that even in high school, middle

school, or elementary school, math classes could follow a very similar path and meet an equal

amount of success. As she states, students are “never too young to experience mathematics in

emotional ways.”81

On the first day of the course, Dr. Cohen begins by reading a story out loud to her

students, easing them into the new class and letting them reflect on what they are hearing. This

activity allows the students to dive into the material without actually demanding very much from

them. The story she favors for this introduction, An Old Arithmetician, written by Mary Eleanor

81 Cohen, “Truth and Beauty,” 534-9.

38

Wilkins Freeman in 1885, deals with an old woman who has a gift of solving “sums.” Of course,

as a work of fiction, this gift also ends up being a curse; the old woman becomes so absorbed

trying to solve a summation problem that her granddaughter goes missing while she is

distracted.82 Using this story as a catalyst, Cohen immediately asks the class who can tell her the

sum of the first 100 integers? The students immediately begin working on the math problem, and

once they have an answer, Cohen provides them with more problems to solve; the infinite sum

1+1/2+1 /4+1 /8+…, and then the more complicated infinite sum 1+1/3+1/9+1/27+… both of

which (as math-inclined students are likely to notice) hint at the geometric series. Then, as a

bonus, for the students who are interested, she mentions that the sum ∑n=1

∞ 1n2 =

π 2

6. This structure

brilliantly introduces the simple idea of repeated sums.83 Everyone already has an understanding

of addition, and these summations are merely the repetition of this basic arithmetic idea. Even

when Cohen introduces infinite sums, which may be difficult to solve, the idea is still incredibly

graspable. It is also important, of course, to note the entire discussion originated out of a fictional

story.

With some of her other assigned stories, Dr. Cohen has addressed topics including

probability, logic, the Pythagorean theorem, modular arithmetic, all the way up through a brief

mentioning of Godel’s Theorem about the nonexistence of complete axiomatic systems. This

theorem states that no set of axioms (an axiom is an argument that is accepted as true without

proof) is sufficient to prove that all facts are true; a fundamental idea in the philosophy of

mathematics. Running into a theorem like this, which is incredibly complicated to prove but

fairly easy to understand, can certainly be a first step into very high-level abstract mathematical

82 Freeman, Mary Eleanor Wilkins. An Old Arithmetician. Charlottesville, Va.: University of Virginia Library, 1995.

83 Cohen, “Truth and Beauty,” 534-9.

39

thinking. Similarly, exposure can lead to an interest in the subject, personal investigations, and

further explorations in related areas of mathematics. If, on the other hand, a student merely finds

the problems boring and monotonous, at the very least they are interacting with the material in a

more exciting manner.

In fact, the culmination of the course, students’ composition of their own work, more

often than not leads to student reflections on their own life-encounters with mathematics. Cohen

tries to keep this conversation, regarding math in the real-world, especially relevant to her

students’ lives throughout the semester. For example, she asks students about family members

who are math enthusiasts, or to remember their favorite math teachers in high school. To keep

things light, she starts with “life questions” before making the transition back to mathematics in

the students’ lives. As one student explained, “in most courses there’s just one day when the

teacher asks us to talk about ourselves, but by then we’re so burned out… we just don’t want to.

But [she does] that throughout the semester, and [she’s] gotten students who normally don’t talk

much to say things in class.”84 Cohen adds, they don’t just talk about anything, but math things.

Most college math professors are likely to scoff at the suggestion of incorporating an

element of reading to their syllabus. Bringing literature into a class whose focus is on an obscure

concept, such as number theory, would take up valuable class time; especially since every field

of mathematics contains an endless number of rabbit holes for further investigation. Why waste

time putting effort into a completely different subject altogether? This argument is valid since

there is so much information to learn, and so little time to learn it. However, the fact that there

are so many areas for further investigation leads to a perfect argument for bringing in literature.

Students could find topics they wouldn’t normally encounter, or run into questions which pique

their curiosity, on their own. Similarly, bringing in related books don’t necessarily have to take

84 Cohen, “Truth and Beauty,” 534-9.

40

up as much class time as it did in Cohen’s course. Reading could be an activity to take place

outside of class and be based on student’s own interests. Because there are so many examples of

mathematical literature out there, students could pursue something that they are interested in,

greatly enhancing their experience with the material. Just as middle school teachers are pushing

for interdisciplinary studies by bringing literature into the classroom, so too could college

professors. Even if minimal amount of class time was spent on the actual books, I believe their

mere introduction could still be greatly beneficial.

David Foster Wallace’s classification of the sub-genre, “math-melodrama,” is just one

example of an additional resource that could help students take more out of the class. The rise in

the popularity, and abundance, of math-inspired media shows that the demand for the material

already exists.85 The success of books like Fermat’s Last Theorem, by Amir Arzel, or even A

Beautiful Mind by Sylvia Nasar further exemplify that people are better relating to the plight of

the mathematician, so why not take advantage of it? The “math-melodrama,” as described by

Wallace, is a sub-genre often depicting the life of a Prometheus-like character who views pure

math as a “mortal quest for divine truth.” This new perception and handling of math could help

inspire students to change their view of the subject. Even in a lower level course, the portrayal of

this eventual goal could give a student a new respect for the subject matter. Then, in an advanced

course, it would help student’s appreciate the material they are about to encounter. As David

Foster Wallace puts it, these books could “bring the subject to life and demonstrate its beauty

and passion. Both readers and math itself stand to gain” (Wallace).

While not all of these books are exactly successful at synthesizing the two subjects, as it

can be difficult, the more mathematical knowledge a student has, the more they “stand to gain”

from reading the book. David Foster Wallace notes the paradox that with many books, the

85 Wallace, “Math Melodrama.” 2263-2267

41

authors are most likely gearing their texts towards those who would already enjoy the

mathematics. Unfortunately, these people are also the most likely to be disappointed by the way

in which the math concepts are covered. Typically the subject ends up being glossed over,

handled vaguely, or overtly imaginary (which doesn’t have to be a problem, as long at the

writing is good enough to still inspire the same message). However, these novels all still have

something to offer. Even if the math concepts are glossed over, a student could likely gain a

desire to know more; to further their understanding of what is taking place in the book, and their

math classroom is the perfect environment for initial inquiry. Thus, when students read these

types of novels while learning about the concepts, they will not only better enjoy the book, but

will also begin to change their perception of math due to its positive portrayal in the book.

Take, for example, the book White Light by Rudy Rucker. Rucker was a professor at the

State Colleges in New York during the 70s where he did research on Cantor’s continuum

hypothesis (to oversimplify, he essentially dealt with the different sizes of infinities). His

research inspired his novel whose protagonist struggles with trying to solve the exact same

problem.86 The notion of infinity becomes a major thread throughout this book, as it brings up

many interesting properties and hypothetical scenarios, taking advantage of the abstract nature of

infinity. For students about to enter a course on set theory, or any other course dealing with

infinity, many ideas presented in the novel are sure to baffle and inspire them. For example, at

one point in the novel, after the protagonist, Felix, has already managed to travel to infinity in

four hours through an infinite acceleration (one billion miles in 2 hours, then the next billion in

one hour, then the next billion in half hour, then ¼ hour, etc., which places them an infinite

distance from earth in four hours), they find themselves at “alef-null,” the most basic, aka

countable, form of infinity.87

86 Rucker, Rudy v. B.. White light. New York: Four Walls Eight Windows, 2001.87 Ibid

42

Already this idea is a bit paradoxical, traveling an infinite distance in a finite amount of

time, but when they reach Hilbert’s Hotel, a famous mathematical paradox, even more fun with

infinity begins. This infinite hotel is already full when Felix arrives, so the question is posed,

how can they fit this new guest? Since the hotel in infinite, Felix figures out the solution: have

every inhabitant move one room over. This means the person in room #1 moves to room #2, #2

moves to room #3, etc., leaving room #1 completely open. A little bit later, the hotel has an

infinite number of guests arrive, all wanting to enter the full hotel, and again Felix has an answer

to fit them all in: have each current guest double their room number, and move to that new room

instead. So room #1 will move to #2, #2 moves to number 4, #3 moves to #6, and so on. This

leaves all the odd room numbers empty, now allowing an infinite number of guests to check into

the hotel.88 This paradox demonstrates that when dealing with the infinite, our notion of a “full”

hotel and a hotel with “no available rooms” are no longer identical. Not only can one new guest

check in, but also the hotel can make room for an infinite number of new inhabitants, something

only possible through the nature of infinity.

Assigning a reading like this outside of class, even as early as the first week, could very

well inspire students’ curiosity about the subject. What are the mathematical principles that make

these events possible? What other properties of infinity exist? Students who have a solid

understanding of math are more likely to find the enjoyment from this type of novel, as the

“magic” of infinity seems to be present in nearly all mathematical fields and the more one learns

about it, the more abstract and interesting this concept becomes. Just as earlier levels of

mathematics utilize literature to help inspire learning, pique the curiosities of students, and

dangle interesting problems in front of them, pure math classes can use reading for the same

goals. True, more class time should probably be spent on the actual material itself, but an outside

88 Rucker, White light.

43

assignment exploring some of these books could very well heighten a student’s experience in the

class. Even if the subject matter is already exciting, students rely solely on their teachers to

present them with thought-provoking ideas. Teachers and professors ultimately have the final say

on the topics being covered, and students inevitably obey and follow. Assignments completed

outside of class, and exposure to new ideas, especially when presented through literature, can be

a great inspiration to students as new ideas rear their heads. Even if the works are fictional, they

can help paint a better picture of what a true mathematician actually does, as opposed to the

commonly held belief that they merely do what most people think “math” is made up of:

formulas and calculations.

Much of the United States is guilty of continuing to further the belief that math is

inherently difficult, and accepting math illiteracy to prosper. Many students struggle to find the

necessity of learning math skills as technology makes calculations available at the click of a

mouse. But there is much more to math than mere arithmetic. Mathematics should be about

problem solving, critical thinking, and being creative; not about formulas, equations, and

number-crunching. This view is too often lost because students are not exposed to the real world

applications of their learning. By showing them how crucial this information will be to their

daily lives, students should, if nothing else, begin to see how important mathematics can be. The

government continually pushes for students to study math and sciences, complaining that we are

not the top performing country in these disciplines. Rather than trying to force students to pursue

something they are not passionate about, I propose teachers should instead try to cultivate this

passion for mathematics in students – leading to an overall improvement in student learning.

Though it may seem counterintuitive, by bringing literature into mathematics classrooms, at all

levels of education, students should begin to gain an appreciation for the subject that they too-

44

often learn to despise. If we can once again present mathematics in an interesting light, the result

will be a perception change in society and better mathematical understanding in students. No one

should plead that they have a math-deficiency. The sooner we begin integrating literature into

math classrooms, the sooner we will begin cultivating student interest and promoting better

understanding of this fascinating and indispensable subject.

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