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This article gives techniques and tips for college mathematics instructors to increase students’ ability to read and comprehend mathematics. The article also includes some relevant history of reading instruction and some motivation for incorporating these ideas into courses.

1.
Journal of Humanistic Mathematics

Volume 9 | Issue 1 January 2019

Preparing Our Students to Read and Understand Mathematics

Melanie Butler

Mount St. Mary's University

Follow this and additional works at: https://scholarship.claremont.edu/jhm

Part of the Arts and Humanities Commons, and the Mathematics Commons

Recommended Citation

Butler, M. "Preparing Our Students to Read and Understand Mathematics," Journal of Humanistic

Mathematics, Volume 9 Issue 1 (January 2019), pages 158-177. DOI: 10.5642/jhummath.201901.08 .

Available at: https://scholarship.claremont.edu/jhm/vol9/iss1/8

©2019 by the authors. This work is licensed under a Creative Commons License.

JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and

published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/

The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds

professional ethical guidelines. However the views and opinions expressed in each published manuscript belong

exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for

them. See https://scholarship.claremont.edu/jhm/policies.html for more information.

Volume 9 | Issue 1 January 2019

Preparing Our Students to Read and Understand Mathematics

Melanie Butler

Mount St. Mary's University

Follow this and additional works at: https://scholarship.claremont.edu/jhm

Part of the Arts and Humanities Commons, and the Mathematics Commons

Recommended Citation

Butler, M. "Preparing Our Students to Read and Understand Mathematics," Journal of Humanistic

Mathematics, Volume 9 Issue 1 (January 2019), pages 158-177. DOI: 10.5642/jhummath.201901.08 .

Available at: https://scholarship.claremont.edu/jhm/vol9/iss1/8

©2019 by the authors. This work is licensed under a Creative Commons License.

JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and

published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/

The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds

professional ethical guidelines. However the views and opinions expressed in each published manuscript belong

exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for

them. See https://scholarship.claremont.edu/jhm/policies.html for more information.

2.
Preparing Our Students to Read and Understand

Mathematics

Melanie Butler

Department of Mathematics & Computer Science, Mount St. Mary’s University,

Maryland, USA

mbutler@msmary.edu

This article gives techniques and tips for college mathematics instructors to in-

crease students’ ability to read and comprehend mathematics. The article also

includes some relevant history of reading instruction and some motivation for

incorporating these ideas into courses.

1. Introduction

We comment that our students don’t use the textbook. Our students com-

plain (on our teaching evaluations at least) that the textbook wasn’t useful.

Pedagogically, we have moved from pure lecture to flipped classrooms, the

Moore method, and other student-centered, constructivist approaches. In ad-

dition, we are being asked more often to think about offering online courses.

We worry about misleading surveys and news and wonder how we can help

our students learn to navigate the large volume of technical information they

come into contact with each day. What is the missing piece that weaves

through all of this? The ability to read and comprehend mathematics.

Several years ago I was given the opportunity to teach a first-year symposium

course at my institution. This course is taught by instructors from different

disciplines, but the focus of the course is on reading and writing. Instructors

for the course undergo intensive summer training. Thus began my endeavors

into teaching reading and comprehension.

Journal of Humanistic Mathematics Volume 9 Number 1 (January 2019)

Mathematics

Melanie Butler

Department of Mathematics & Computer Science, Mount St. Mary’s University,

Maryland, USA

mbutler@msmary.edu

This article gives techniques and tips for college mathematics instructors to in-

crease students’ ability to read and comprehend mathematics. The article also

includes some relevant history of reading instruction and some motivation for

incorporating these ideas into courses.

1. Introduction

We comment that our students don’t use the textbook. Our students com-

plain (on our teaching evaluations at least) that the textbook wasn’t useful.

Pedagogically, we have moved from pure lecture to flipped classrooms, the

Moore method, and other student-centered, constructivist approaches. In ad-

dition, we are being asked more often to think about offering online courses.

We worry about misleading surveys and news and wonder how we can help

our students learn to navigate the large volume of technical information they

come into contact with each day. What is the missing piece that weaves

through all of this? The ability to read and comprehend mathematics.

Several years ago I was given the opportunity to teach a first-year symposium

course at my institution. This course is taught by instructors from different

disciplines, but the focus of the course is on reading and writing. Instructors

for the course undergo intensive summer training. Thus began my endeavors

into teaching reading and comprehension.

Journal of Humanistic Mathematics Volume 9 Number 1 (January 2019)

3.
Melanie Butler 159

I soon realized that the techniques I was learning from faculty across campus

would be valuable when modified for my mathematics courses. I began asking

my mathematics students to read more and started investigating ways to

support them in this reading.

Recently I had the chance to delve even deeper into the subject. During a

sabbatical, I read research literature on literacy, on content area literacy, on

disciplinary literacy, and on teaching reading in mathematics. Unfortunately,

there are not a lot of resources for teaching reading in mathematics at the

college level. Many times I was frustrated by being unable to locate resources

or by resources being out of print or out of date. Many resources dealt only

with teaching students to decode word problems.

Looking for new ideas, I began to interview my colleagues. I talked with

mathematics faculty at my own and other institutions. I interviewed faculty

in foreign languages, philosophy, education, English, history, and theology.

I talked with K-12 educators. I asked all of these teachers how they them-

selves read, how they teach students to read, and how they use activities or

strategies to help students make sense of written material.

It is my goal with this article to bring resources together for college instruc-

tors to incorporate more teaching of reading into college-level mathematics.

This article includes information from my experiences teaching reading gen-

erally and in mathematics, my interviews with teachers and faculty at various

levels and from various disciplines, research literature, and educational re-

sources. This article is meant to give ideas and techniques that you can use

in your mathematics classes, for non-majors and majors alike, right now.

1.1. Motivation

Why should you teach students to read mathematics? In our classrooms,

as we have moved away from lecture as a primary teaching strategy,

we often ask our students to explore mathematics concepts on their own,

through a textbook or online video. We are using more class time for

group work, flipped classrooms, and constructivist activities. We are ask-

ing our students to begin the learning process on their own, without nec-

essarily equipping them with the skills they need to do so. For example,

many students took high school English classes centered on reading literature.

I soon realized that the techniques I was learning from faculty across campus

would be valuable when modified for my mathematics courses. I began asking

my mathematics students to read more and started investigating ways to

support them in this reading.

Recently I had the chance to delve even deeper into the subject. During a

sabbatical, I read research literature on literacy, on content area literacy, on

disciplinary literacy, and on teaching reading in mathematics. Unfortunately,

there are not a lot of resources for teaching reading in mathematics at the

college level. Many times I was frustrated by being unable to locate resources

or by resources being out of print or out of date. Many resources dealt only

with teaching students to decode word problems.

Looking for new ideas, I began to interview my colleagues. I talked with

mathematics faculty at my own and other institutions. I interviewed faculty

in foreign languages, philosophy, education, English, history, and theology.

I talked with K-12 educators. I asked all of these teachers how they them-

selves read, how they teach students to read, and how they use activities or

strategies to help students make sense of written material.

It is my goal with this article to bring resources together for college instruc-

tors to incorporate more teaching of reading into college-level mathematics.

This article includes information from my experiences teaching reading gen-

erally and in mathematics, my interviews with teachers and faculty at various

levels and from various disciplines, research literature, and educational re-

sources. This article is meant to give ideas and techniques that you can use

in your mathematics classes, for non-majors and majors alike, right now.

1.1. Motivation

Why should you teach students to read mathematics? In our classrooms,

as we have moved away from lecture as a primary teaching strategy,

we often ask our students to explore mathematics concepts on their own,

through a textbook or online video. We are using more class time for

group work, flipped classrooms, and constructivist activities. We are ask-

ing our students to begin the learning process on their own, without nec-

essarily equipping them with the skills they need to do so. For example,

many students took high school English classes centered on reading literature.

4.
160 Preparing Our Students to Read and Understand Mathematics

But how many students have had the opportunity to talk about reading a

mathematics text? During one of my interviews, a colleague mentioned that

he didn’t learn how to really read until graduate school and then he did so by

seeing his professors do it. We can equip our students to read mathematics

now if we take the time to do so.

If I can teach my students to read mathematics and make sense of it, then

I have done them a better service than teaching them just the mathematics.

For example, I can teach a student to solve a first-order linear differential

equation, but will she remember it five years from now when she needs to

apply this skill in her job? She is better off if I prepared her to read and

understand the mathematics, and she can look up the skill and refresh herself.

Preservice teachers will also find this skill invaluable. As teachers we all come

across something we need to teach that we have never seen before. The ability

to read about it and comprehend it is vitally important.

There are several calls for mathematicians to include more reading instruc-

tion in college-level mathematics. In A Common Vision for Undergraduate

Mathematical Sciences Programs in 2025, Karen Saxe and Linda Braddy [25]

coalesce the recommendations from seven curricular guides published by five

professional associations into a set of principles to guide mathematics higher

education. All of the guides recommend that, “Instructors should intention-

ally plan curricula toimprove students’ ability to communicate quantitative

ideas orally and in writing (and since a precursor to communication is un-

derstanding, improve students’ ability to interpret information, organize ma-

terial, and reflect on results)” [25, page 13]. In addition, they found that

pedagogy that lets students be actively involved in reading, synthesizing,

and evaluating course content is recommended frequently from these organi-

zations [25, page 19].

Other sources echo these calls. Fang and Coatoam [10] state that, “de-

veloping disciplinary literacy is a long-term process that begins in upper-

elementary grades and continues through college.” The NCTM Principals

and Standards for School Mathematics [22, page 60] states that, “students

who have opportunities, encouragement, and support for writing, reading,

and listening in mathematics classes reap dual benefits: they communi-

cate to learn mathematics, and they learn to communicate mathematically.”

But how many students have had the opportunity to talk about reading a

mathematics text? During one of my interviews, a colleague mentioned that

he didn’t learn how to really read until graduate school and then he did so by

seeing his professors do it. We can equip our students to read mathematics

now if we take the time to do so.

If I can teach my students to read mathematics and make sense of it, then

I have done them a better service than teaching them just the mathematics.

For example, I can teach a student to solve a first-order linear differential

equation, but will she remember it five years from now when she needs to

apply this skill in her job? She is better off if I prepared her to read and

understand the mathematics, and she can look up the skill and refresh herself.

Preservice teachers will also find this skill invaluable. As teachers we all come

across something we need to teach that we have never seen before. The ability

to read about it and comprehend it is vitally important.

There are several calls for mathematicians to include more reading instruc-

tion in college-level mathematics. In A Common Vision for Undergraduate

Mathematical Sciences Programs in 2025, Karen Saxe and Linda Braddy [25]

coalesce the recommendations from seven curricular guides published by five

professional associations into a set of principles to guide mathematics higher

education. All of the guides recommend that, “Instructors should intention-

ally plan curricula toimprove students’ ability to communicate quantitative

ideas orally and in writing (and since a precursor to communication is un-

derstanding, improve students’ ability to interpret information, organize ma-

terial, and reflect on results)” [25, page 13]. In addition, they found that

pedagogy that lets students be actively involved in reading, synthesizing,

and evaluating course content is recommended frequently from these organi-

zations [25, page 19].

Other sources echo these calls. Fang and Coatoam [10] state that, “de-

veloping disciplinary literacy is a long-term process that begins in upper-

elementary grades and continues through college.” The NCTM Principals

and Standards for School Mathematics [22, page 60] states that, “students

who have opportunities, encouragement, and support for writing, reading,

and listening in mathematics classes reap dual benefits: they communi-

cate to learn mathematics, and they learn to communicate mathematically.”

5.
Melanie Butler 161

Finally, in Literacy Strategies for Improving Mathematics Instruction, Joan

Kenney [17, page 11] states that, “If we intend for students to understand

mathematical concepts rather than to produce specific performances, we

must teach them to engage meaningfully with mathematics texts.” While

Kenney and the NCTM may have been speaking to K-12 educators, higher

education needs to attend to reading mathematics as well.

1.2. History

There are many books and articles that provide a history of literacy educa-

tion, of constructivist learning theories, and of other modern movements in

education. One recommendation for a general overview is Chapter 1 of the

2013 book Theoretical Models and Processes of Reading by Reynold et al.,

where Alexander and Fox summarize theories of reading over the last cen-

tury, including historical and political perspectives [4]. For this section, I am

mainly concerned with current theories and understanding of reading, so I

note the relevant, recent ideas. My goal is to provide a brief recent historical

account to motivate and support some of the strategies and activities.

The importance of motivating students to read emerged in the early 2000s

[14]. The importance of motivation coincides with the idea that readers don’t

just passively absorb the material; there is more of a transaction between the

reading and the reader, as well as a requirement that the reader participate

actively when reading. The transactional and motivational theories of read-

ing are a lot like the constructivist theory of learning. In addition, during the

early 2000s, educators and researchers began to pay more attention to the

fact that growing as a reader is important for people of all ages and abilities

[2]. Educators and researchers also began to study the role of alternative

forms of text in reading comprehension as technology changes the way we

encounter written material [27].

In the next decade, researchers began to explore what it means for

students to develop a deeper understanding of a text more closely tied

to critical thinking and taxonomies of thinking. Kulikowich and Alexan-

der [18] extended the idea of being an engaged reader to being engaged

with higher-order thinking goals related to critical analysis and evaluation.

Murphy et al. [21, page 741] in their meta-analysis defined critical liter-

acy as “higher order thinking and critical reflection on text and discourse”.

Finally, in Literacy Strategies for Improving Mathematics Instruction, Joan

Kenney [17, page 11] states that, “If we intend for students to understand

mathematical concepts rather than to produce specific performances, we

must teach them to engage meaningfully with mathematics texts.” While

Kenney and the NCTM may have been speaking to K-12 educators, higher

education needs to attend to reading mathematics as well.

1.2. History

There are many books and articles that provide a history of literacy educa-

tion, of constructivist learning theories, and of other modern movements in

education. One recommendation for a general overview is Chapter 1 of the

2013 book Theoretical Models and Processes of Reading by Reynold et al.,

where Alexander and Fox summarize theories of reading over the last cen-

tury, including historical and political perspectives [4]. For this section, I am

mainly concerned with current theories and understanding of reading, so I

note the relevant, recent ideas. My goal is to provide a brief recent historical

account to motivate and support some of the strategies and activities.

The importance of motivating students to read emerged in the early 2000s

[14]. The importance of motivation coincides with the idea that readers don’t

just passively absorb the material; there is more of a transaction between the

reading and the reader, as well as a requirement that the reader participate

actively when reading. The transactional and motivational theories of read-

ing are a lot like the constructivist theory of learning. In addition, during the

early 2000s, educators and researchers began to pay more attention to the

fact that growing as a reader is important for people of all ages and abilities

[2]. Educators and researchers also began to study the role of alternative

forms of text in reading comprehension as technology changes the way we

encounter written material [27].

In the next decade, researchers began to explore what it means for

students to develop a deeper understanding of a text more closely tied

to critical thinking and taxonomies of thinking. Kulikowich and Alexan-

der [18] extended the idea of being an engaged reader to being engaged

with higher-order thinking goals related to critical analysis and evaluation.

Murphy et al. [21, page 741] in their meta-analysis defined critical liter-

acy as “higher order thinking and critical reflection on text and discourse”.

6.
162 Preparing Our Students to Read and Understand Mathematics

By this definition, the authors suggest that students should be able to go

beyond a basic reading and comprehension of texts and instead should be

taught to think critically and reflectively about what they are reading. To

achieve this goal, Murphy et al. [21] encourage teachers to have students do

more than offer opinions on a text; rather they suggest students be taught

and challenged to defend and support what they say about a text or how

they interpret a text. Furthermore, the authors believe this behavior needs

to be modeled by a competent person.

Other researchers studied the importance of disciplinary differences between

texts and what it means to teach students to read with comprehension in

different disciplines, such as science [19]. Alexander [3] introduced a Model

of Domain Learning (MDL), which is applicable to many traditional school

subjects and to mathematics, in particular. In this model, development

of expertise in a particular domain is broken down into three categories:

knowledge, interest, and strategic processing. Knowledge is further broken

down into breadth of knowledge across the domain and depth of knowledge

on particular topics in the domain. Interest is measured in terms of interest

in a particular situation, such as a student’s interest in learning about limits

at infinity while reading a Calculus book on the topic, and in terms of the

overall individual level of interest in a domain. Finally, strategic processing

centers on strategies the individual employs while reading in the particular

domain, such as summarizing and self-evaluation. Furthermore, Alexander

[3] describes surface-level reading strategies, such as rereading, that help a

student gain basic understanding of a text. Deep-level processing strategies,

like relating a topic to prior knowledge, are employed by individuals who can

convert what they have read into their own message.

Recently there have been more studies into reading mathematics at the

college-level. Weinberg et al. [29] completed a study of 1156 undergradu-

ates in introductory mathematics classes in an effort to understand how

students use their math textbooks. The researchers found that students of-

ten use examples, rather than the written explanations, to help them un-

derstand a concept and noted that this tendency could be problematic.

They suggest that mathematics faculty should ask students to read and

should provide instruction on how to do so. Weber [28] researched how

successful math students read mathematical proofs. From videotaped ses-

sions with four undergraduates, Weber found four typically used strategies:

By this definition, the authors suggest that students should be able to go

beyond a basic reading and comprehension of texts and instead should be

taught to think critically and reflectively about what they are reading. To

achieve this goal, Murphy et al. [21] encourage teachers to have students do

more than offer opinions on a text; rather they suggest students be taught

and challenged to defend and support what they say about a text or how

they interpret a text. Furthermore, the authors believe this behavior needs

to be modeled by a competent person.

Other researchers studied the importance of disciplinary differences between

texts and what it means to teach students to read with comprehension in

different disciplines, such as science [19]. Alexander [3] introduced a Model

of Domain Learning (MDL), which is applicable to many traditional school

subjects and to mathematics, in particular. In this model, development

of expertise in a particular domain is broken down into three categories:

knowledge, interest, and strategic processing. Knowledge is further broken

down into breadth of knowledge across the domain and depth of knowledge

on particular topics in the domain. Interest is measured in terms of interest

in a particular situation, such as a student’s interest in learning about limits

at infinity while reading a Calculus book on the topic, and in terms of the

overall individual level of interest in a domain. Finally, strategic processing

centers on strategies the individual employs while reading in the particular

domain, such as summarizing and self-evaluation. Furthermore, Alexander

[3] describes surface-level reading strategies, such as rereading, that help a

student gain basic understanding of a text. Deep-level processing strategies,

like relating a topic to prior knowledge, are employed by individuals who can

convert what they have read into their own message.

Recently there have been more studies into reading mathematics at the

college-level. Weinberg et al. [29] completed a study of 1156 undergradu-

ates in introductory mathematics classes in an effort to understand how

students use their math textbooks. The researchers found that students of-

ten use examples, rather than the written explanations, to help them un-

derstand a concept and noted that this tendency could be problematic.

They suggest that mathematics faculty should ask students to read and

should provide instruction on how to do so. Weber [28] researched how

successful math students read mathematical proofs. From videotaped ses-

sions with four undergraduates, Weber found four typically used strategies:

7.
Melanie Butler 163

Try to do it yourself, identify the proof framework, break the proof into

smaller subproofs, and use examples to understand difficult parts. Carducci

[8] has used technical instructions, such as instructions on how to do a com-

plex card trick, to help students reflect on how they read the instructions.

Students are then asked to translate these same skills into reading mathe-

1.3. Article Organization

The next two sections are divided into strategies and activities for planning

a course that incorporates instruction on reading mathematics (Section 2)

and strategies and activities to prepare students before an assigned reading

(Section 3). Section 4 includes a short conclusion. A follow-up article is

planned on strategies and activities during reading and post reading. In

some cases, a strategy may be used at more than one stage. In addition,

some strategies may have parts that extend into multiple stages. Each of

these stages is important for comprehension. Strategies are described as for

an individual, for a small group, or for the whole class; however, many can

be modified to be used in one of the other ways.

The strategies for pre-reading help prepare and motivate students for the

reading assignment. In both the research literature I have explored and

in interviews I had with faculty, it was apparent to me that reading with a

purpose that students understand is very important. If we think of learning as

being goal-directed, then we have to think about what our students interpret

as the goals of our reading assignments.

Finally, you might consider using some of these strategies for helping students

make sense of other types of materials, such as online videos. Unfortunately,

research on the topic is limited, so careful reflection is needed on what would

be appropriate and helpful to the students. Modifications may need to be

made to make the activities appropriate for other types of resources.

2. Designing Your Course

There are many strategies and activities you can incorporate into your course

while it is in progress. However, if you have the luxury of planning ahead,

there are some things to consider and some things you may want to plan in

your syllabus and course.

Try to do it yourself, identify the proof framework, break the proof into

smaller subproofs, and use examples to understand difficult parts. Carducci

[8] has used technical instructions, such as instructions on how to do a com-

plex card trick, to help students reflect on how they read the instructions.

Students are then asked to translate these same skills into reading mathe-

1.3. Article Organization

The next two sections are divided into strategies and activities for planning

a course that incorporates instruction on reading mathematics (Section 2)

and strategies and activities to prepare students before an assigned reading

(Section 3). Section 4 includes a short conclusion. A follow-up article is

planned on strategies and activities during reading and post reading. In

some cases, a strategy may be used at more than one stage. In addition,

some strategies may have parts that extend into multiple stages. Each of

these stages is important for comprehension. Strategies are described as for

an individual, for a small group, or for the whole class; however, many can

be modified to be used in one of the other ways.

The strategies for pre-reading help prepare and motivate students for the

reading assignment. In both the research literature I have explored and

in interviews I had with faculty, it was apparent to me that reading with a

purpose that students understand is very important. If we think of learning as

being goal-directed, then we have to think about what our students interpret

as the goals of our reading assignments.

Finally, you might consider using some of these strategies for helping students

make sense of other types of materials, such as online videos. Unfortunately,

research on the topic is limited, so careful reflection is needed on what would

be appropriate and helpful to the students. Modifications may need to be

made to make the activities appropriate for other types of resources.

2. Designing Your Course

There are many strategies and activities you can incorporate into your course

while it is in progress. However, if you have the luxury of planning ahead,

there are some things to consider and some things you may want to plan in

your syllabus and course.

8.
164 Preparing Our Students to Read and Understand Mathematics

2.1. Textbooks

If you plan to assign readings from the textbook, pick your text very care-

fully. Many textbooks are written in a way that is difficult to understand

for students that are first learning a subject. Since motivation is often im-

portant in getting students to read, a textbook that is too hard can be very

discouraging. Look for texts that are meant for students to read. Bullock

and Millman [6] studied how mathematicians write versus how students read.

They found that mathematicians value brevity and conciseness in mathemat-

ics. However, students who are reading mathematics are not best-served by

these traits. Instead of following the conventions of mathematics, Bullock

and Millman [6] argue that texts meant for students to read should meet

the needs of the student. In particular, if you plan to ask your students to

read from their textbook, look for textbooks that incorporate more exposi-

tory writing, more examples, and mathematics that is not always what we

think of as the “best” (e.g., shortest possible proof of a theorem or calculus

example worked out with all the algebra steps skipped), but will instead give

students the information they need to process the text.

Some faculty that assign readings from the textbook suggest using a textbook

with short sections or chapters. These faculty note that students find the

shorter sections less intimidating. Other faculty who I interviewed empha-

sized the need for students to write in texts as they read; so they discourage

their students from renting books or using ebooks. Students may be able to

use software or buy ebooks where they can take notes in the ebook.

We’ve all encountered bad texts. Even in texts that we like, there are often

elements that we don’t like. Sometimes we disagree about what is good or

bad in a text. The point here is that the authors of these texts are human,

they make mistakes, and they have to make decisions about how to present

the ideas in their texts; we may not always agree with these decisions. Help

your students to see that if they can’t understand something in a text, the

author may have done a bad job explaining it!

2.2. Other Readings

Farmer and Schielak [11] argue that other types of readings, besides text-

books, are very important to include in mathematics classrooms. They cite

the need to change attitudes toward mathematics as a primary reason.

2.1. Textbooks

If you plan to assign readings from the textbook, pick your text very care-

fully. Many textbooks are written in a way that is difficult to understand

for students that are first learning a subject. Since motivation is often im-

portant in getting students to read, a textbook that is too hard can be very

discouraging. Look for texts that are meant for students to read. Bullock

and Millman [6] studied how mathematicians write versus how students read.

They found that mathematicians value brevity and conciseness in mathemat-

ics. However, students who are reading mathematics are not best-served by

these traits. Instead of following the conventions of mathematics, Bullock

and Millman [6] argue that texts meant for students to read should meet

the needs of the student. In particular, if you plan to ask your students to

read from their textbook, look for textbooks that incorporate more exposi-

tory writing, more examples, and mathematics that is not always what we

think of as the “best” (e.g., shortest possible proof of a theorem or calculus

example worked out with all the algebra steps skipped), but will instead give

students the information they need to process the text.

Some faculty that assign readings from the textbook suggest using a textbook

with short sections or chapters. These faculty note that students find the

shorter sections less intimidating. Other faculty who I interviewed empha-

sized the need for students to write in texts as they read; so they discourage

their students from renting books or using ebooks. Students may be able to

use software or buy ebooks where they can take notes in the ebook.

We’ve all encountered bad texts. Even in texts that we like, there are often

elements that we don’t like. Sometimes we disagree about what is good or

bad in a text. The point here is that the authors of these texts are human,

they make mistakes, and they have to make decisions about how to present

the ideas in their texts; we may not always agree with these decisions. Help

your students to see that if they can’t understand something in a text, the

author may have done a bad job explaining it!

2.2. Other Readings

Farmer and Schielak [11] argue that other types of readings, besides text-

books, are very important to include in mathematics classrooms. They cite

the need to change attitudes toward mathematics as a primary reason.

9.
Melanie Butler 165

They also suggest that readings on recreational mathematics can help stu-

dents see the beauty and fun of math, which can lead to an increased interest

in the subject. Thus, even if you are assigning reading from the textbook,

consider incorporating other types of readings. These other readings might

include history, expository writing, newspaper articles, research articles, ar-

ticles for a wider audience, and fiction. For ideas, consider the Mathemat-

ical Reading List [30] made available online in 2015 by the University of

Cambridge Mathematics Faculty for their students. The list is broken into

categories like history, recreational, and readable mathematics. Each sugges-

tion includes a description. Farmer and Schielak [11] also provide a reading

list, broken down by topic, as well as sample study guides for readings. In

addition, Karaali [15] suggests using philosophical readings in mathematical

courses. Specifically she provides a list of philosophical readings to be used

in a linear algebra course, with the goal of helping students to see why the

mathematics they are learning is relevant to the world around us.

In research literature, education literature, and interviews, I heard many

bring up the importance of including history in the teaching of mathematics.

In interviews, faculty mentioned that mathematics is a human enterprise,

but that it is not often seen this way by students because of the dry way

it can be presented in textbooks or in lectures. Sharing readings on the

history of mathematics with students will help them to see the human side

of mathematics and to see that throughout history humans have struggled

with mathematical ideas. In addition, during interviews, faculty pointed out

that history can help provide a context for the mathematics content and

the vocabulary. An English teacher introducing a Victorian-era novel to her

students would provide context in terms of the history of the time and life

of the author; such steps could also be useful in mathematics.

Grabiner [13] gives three reasons for including history in the teaching of

mathematics. She notes that an historical perspective can help students to

see the inherent difficulty of some mathematical concepts. Secondly, she sug-

gests that seeing the historical development of mathematics can motivate

students to study mathematics. Finally, Grabiner [13] believes history can

help tie mathematics into the greater tradition of human thinking and ad-

vancements. Byers [7] gives a history of including mathematics history in

mathematics classes.

They also suggest that readings on recreational mathematics can help stu-

dents see the beauty and fun of math, which can lead to an increased interest

in the subject. Thus, even if you are assigning reading from the textbook,

consider incorporating other types of readings. These other readings might

include history, expository writing, newspaper articles, research articles, ar-

ticles for a wider audience, and fiction. For ideas, consider the Mathemat-

ical Reading List [30] made available online in 2015 by the University of

Cambridge Mathematics Faculty for their students. The list is broken into

categories like history, recreational, and readable mathematics. Each sugges-

tion includes a description. Farmer and Schielak [11] also provide a reading

list, broken down by topic, as well as sample study guides for readings. In

addition, Karaali [15] suggests using philosophical readings in mathematical

courses. Specifically she provides a list of philosophical readings to be used

in a linear algebra course, with the goal of helping students to see why the

mathematics they are learning is relevant to the world around us.

In research literature, education literature, and interviews, I heard many

bring up the importance of including history in the teaching of mathematics.

In interviews, faculty mentioned that mathematics is a human enterprise,

but that it is not often seen this way by students because of the dry way

it can be presented in textbooks or in lectures. Sharing readings on the

history of mathematics with students will help them to see the human side

of mathematics and to see that throughout history humans have struggled

with mathematical ideas. In addition, during interviews, faculty pointed out

that history can help provide a context for the mathematics content and

the vocabulary. An English teacher introducing a Victorian-era novel to her

students would provide context in terms of the history of the time and life

of the author; such steps could also be useful in mathematics.

Grabiner [13] gives three reasons for including history in the teaching of

mathematics. She notes that an historical perspective can help students to

see the inherent difficulty of some mathematical concepts. Secondly, she sug-

gests that seeing the historical development of mathematics can motivate

students to study mathematics. Finally, Grabiner [13] believes history can

help tie mathematics into the greater tradition of human thinking and ad-

vancements. Byers [7] gives a history of including mathematics history in

mathematics classes.

10.
166 Preparing Our Students to Read and Understand Mathematics

In addition to different types of readings, you might have students read on

the same topic from several sources. Think of each text as a teacher — some

are better than others. No one resource is going to give you everything you

need to understand something. By reading multiple sources on the same

topic, the reader has to consider the same idea from multiple perspectives

and can develop a deeper understanding. As instructors we might do this

when we are preparing a class on a topic (we might look at the same topic

in multiple textbooks), but our students can benefit from this repetition in

a similar way.

2.3. Class Time

As educators adjust to new pedagogical methods, they occasionally struggle

to find class time to “give up” to devote to group work, hands-on activities,

or flipped classrooms. However, when used well, these activities enhance stu-

dent learning, rather than detract from it. In the same way, devoting class

time to comprehension of mathematics texts has benefits that outweigh the

costs of “giving up” more class time. Many of the comprehension strate-

gies involve rich, student-centered discussion, which allow students to make

meaning of the written mathematics. Often the teacher does not lead the

class, but rather is a part of the classroom conversation.

By devoting class time to an activity and emphasizing it through course

policies and assignments, we show students what we think it important.

By devoting class time to the comprehension of written mathematics, we

communicate to students that we think it is important. Consider using class

time to read mathematics. There are strategies for getting students to read in

pairs, but students could also read out loud or quietly on their own. Letting

students read in class with other students and you there for support is like

a flipped classroom for reading. Letting students read out loud helps them

to slow down, focus, and hear things in a different way. In addition, having

students read out loud means that all students have heard them same thing,

so can have a democratizing effect on the class.

If you plan to assign readings from a text, take the time in class to orient

students to the text. Let them explore the text and ask questions. Talk

about special features the text might have. Have the students get the text

out, locate things in the text, and get used to it as a classroom resource.

In addition to different types of readings, you might have students read on

the same topic from several sources. Think of each text as a teacher — some

are better than others. No one resource is going to give you everything you

need to understand something. By reading multiple sources on the same

topic, the reader has to consider the same idea from multiple perspectives

and can develop a deeper understanding. As instructors we might do this

when we are preparing a class on a topic (we might look at the same topic

in multiple textbooks), but our students can benefit from this repetition in

a similar way.

2.3. Class Time

As educators adjust to new pedagogical methods, they occasionally struggle

to find class time to “give up” to devote to group work, hands-on activities,

or flipped classrooms. However, when used well, these activities enhance stu-

dent learning, rather than detract from it. In the same way, devoting class

time to comprehension of mathematics texts has benefits that outweigh the

costs of “giving up” more class time. Many of the comprehension strate-

gies involve rich, student-centered discussion, which allow students to make

meaning of the written mathematics. Often the teacher does not lead the

class, but rather is a part of the classroom conversation.

By devoting class time to an activity and emphasizing it through course

policies and assignments, we show students what we think it important.

By devoting class time to the comprehension of written mathematics, we

communicate to students that we think it is important. Consider using class

time to read mathematics. There are strategies for getting students to read in

pairs, but students could also read out loud or quietly on their own. Letting

students read in class with other students and you there for support is like

a flipped classroom for reading. Letting students read out loud helps them

to slow down, focus, and hear things in a different way. In addition, having

students read out loud means that all students have heard them same thing,

so can have a democratizing effect on the class.

If you plan to assign readings from a text, take the time in class to orient

students to the text. Let them explore the text and ask questions. Talk

about special features the text might have. Have the students get the text

out, locate things in the text, and get used to it as a classroom resource.

11.
Melanie Butler 167

Let the students see you use the text and reference it. During interviews,

several faculty mentioned having the students access the text during class

and frequently have the text in their hands.

2.4. Other Considerations

By helping students to feel comfortable reading mathematics, you also help

them to see that, as their instructor, you are not the only source of knowledge

on the topic. Students should be given the ability to find answers, additional

information, and perspectives from other sources.

Consider assigning students to read the same thing from the same source

more than once, leaving time between the two readings. Many faculty across

disciplines read material more than once, but students may not understand

how important this is. Even if we suggest to students that they read some-

thing more than once, they may not do it because they do not understand

why it is important. If you can incorporate two assignments of the same

reading with clear goals, students will begin to see the value in rereading.

We often assess comprehension, but, in a very important study, Dolores

Durkin [9] found that we are not teaching students how to comprehend.

Strategies in this article can help make actively engaging with written mate-

rial part of learning how to comprehend mathematics.

In one interview, a foreign language professor mentioned that when teaching

reading in a foreign language, there is non-native empathy: the idea that,

for example, a Spanish teacher that is a non-native Spanish speaker has

gone through the same thing that the students are going through. In some

sense, we are all non-native to the vocabulary and notation that we use in

mathematics. Let the students know that you have struggled and can still

struggle with reading mathematics.

3. Pre Reading

3.1. Vocabulary and Notation Instruction

Before being assigned a reading students often need exposure to new

vocabulary and notation. McKeown, Beck, and Blake [20] declare that

for vocabulary instruction to be helpful, it needs to be meaningful;

Let the students see you use the text and reference it. During interviews,

several faculty mentioned having the students access the text during class

and frequently have the text in their hands.

2.4. Other Considerations

By helping students to feel comfortable reading mathematics, you also help

them to see that, as their instructor, you are not the only source of knowledge

on the topic. Students should be given the ability to find answers, additional

information, and perspectives from other sources.

Consider assigning students to read the same thing from the same source

more than once, leaving time between the two readings. Many faculty across

disciplines read material more than once, but students may not understand

how important this is. Even if we suggest to students that they read some-

thing more than once, they may not do it because they do not understand

why it is important. If you can incorporate two assignments of the same

reading with clear goals, students will begin to see the value in rereading.

We often assess comprehension, but, in a very important study, Dolores

Durkin [9] found that we are not teaching students how to comprehend.

Strategies in this article can help make actively engaging with written mate-

rial part of learning how to comprehend mathematics.

In one interview, a foreign language professor mentioned that when teaching

reading in a foreign language, there is non-native empathy: the idea that,

for example, a Spanish teacher that is a non-native Spanish speaker has

gone through the same thing that the students are going through. In some

sense, we are all non-native to the vocabulary and notation that we use in

mathematics. Let the students know that you have struggled and can still

struggle with reading mathematics.

3. Pre Reading

3.1. Vocabulary and Notation Instruction

Before being assigned a reading students often need exposure to new

vocabulary and notation. McKeown, Beck, and Blake [20] declare that

for vocabulary instruction to be helpful, it needs to be meaningful;

12.
168 Preparing Our Students to Read and Understand Mathematics

students need to find ways to make sense of the new words that they will

be able to remember. We want students to connect new vocabulary with

prior knowledge and with associated known words. Here I describe some

techniques that can help do just that. In each of these cases, students may

also benefit from sharing their work in groups or as a class.

There are different models for introducing new vocabulary that help achieve

the goals McKeown, Beck and Blake [20] set forth. One strategy is com-

monly called the Frayer Model [12]. Here students think of four categories:

definition, interesting facts, examples, and non-examples. Sometimes the

categories may have different names such as essential characteristics, non-

essential characteristics, examples, and non-examples. Here is a completed

example of using a Frayer Model to define a subgroup in abstract algebra.

Subgroup

Definition: If (G, ∗) is a Interesting Facts: G is a

group and H is a subset of subgroup of itself.

G, then H is a subgroup of The set consisting of just

G if (H, ∗) is also a group. the identity element with

the operation forms a sub-

group.

Examples: (2Z, +) is a sub- Non-examples: {0, 2, 3} is

group of (Z, +). not a subgroup of Z8 since

{0, 4} is a subgroup of Z8 . 2 + 3 = 5, which is not an

element of the subset.

Another model emphasizes a visual representation, where students think of

the word, the definition, something they associate with the word, and draw

a picture. Here is a completed example of this model to define an open set

in topology.

Word Definition: A set is open if it is a neigh-

Open borhood of every point.

Associated Picture:

Word:

neighborhood

students need to find ways to make sense of the new words that they will

be able to remember. We want students to connect new vocabulary with

prior knowledge and with associated known words. Here I describe some

techniques that can help do just that. In each of these cases, students may

also benefit from sharing their work in groups or as a class.

There are different models for introducing new vocabulary that help achieve

the goals McKeown, Beck and Blake [20] set forth. One strategy is com-

monly called the Frayer Model [12]. Here students think of four categories:

definition, interesting facts, examples, and non-examples. Sometimes the

categories may have different names such as essential characteristics, non-

essential characteristics, examples, and non-examples. Here is a completed

example of using a Frayer Model to define a subgroup in abstract algebra.

Subgroup

Definition: If (G, ∗) is a Interesting Facts: G is a

group and H is a subset of subgroup of itself.

G, then H is a subgroup of The set consisting of just

G if (H, ∗) is also a group. the identity element with

the operation forms a sub-

group.

Examples: (2Z, +) is a sub- Non-examples: {0, 2, 3} is

group of (Z, +). not a subgroup of Z8 since

{0, 4} is a subgroup of Z8 . 2 + 3 = 5, which is not an

element of the subset.

Another model emphasizes a visual representation, where students think of

the word, the definition, something they associate with the word, and draw

a picture. Here is a completed example of this model to define an open set

in topology.

Word Definition: A set is open if it is a neigh-

Open borhood of every point.

Associated Picture:

Word:

neighborhood

13.
Melanie Butler 169

Another model, the feature analysis strategy [16] uses connections between

vocabulary words. Students make notes or highlight important examples in

each box to illustrate whether the words down the left-hand side have the

characteristics along the top. Here is an example from calculus that has not

been completed.

Is always Always Always May have May have

continuous has has removable irremovable

domain range discontinuities discontinuities

all real all real

numbers numbers

constant

function

linear

function

quadratic

function

polynomial

function

rational

function

trig

function

Many words in mathematics have a meaning that is different from their

meaning in everyday language. The Typical to Technical Approach [24]

helps students with these common areas of confusion. In this approach, start

by discussing the usual meaning of the word in everyday language. Then

contrast this with the technical definition of the word. Reinforce the mean-

ings with exercises that require that students differentiate between the two

meanings. As an example, consider the word “solution”. Students sometimes

have trouble understanding what we are asking when we ask students to find

a solution to an equation, i.e., a value of the variable that makes the equation

true. In everyday language we think of a solution as a way to deal with a

difficult situation. We can contrast these typical and technical definitions

with sentences like the following.

• A solution to 2x + 3 = 5 is x = 1.

Another model, the feature analysis strategy [16] uses connections between

vocabulary words. Students make notes or highlight important examples in

each box to illustrate whether the words down the left-hand side have the

characteristics along the top. Here is an example from calculus that has not

been completed.

Is always Always Always May have May have

continuous has has removable irremovable

domain range discontinuities discontinuities

all real all real

numbers numbers

constant

function

linear

function

quadratic

function

polynomial

function

rational

function

trig

function

Many words in mathematics have a meaning that is different from their

meaning in everyday language. The Typical to Technical Approach [24]

helps students with these common areas of confusion. In this approach, start

by discussing the usual meaning of the word in everyday language. Then

contrast this with the technical definition of the word. Reinforce the mean-

ings with exercises that require that students differentiate between the two

meanings. As an example, consider the word “solution”. Students sometimes

have trouble understanding what we are asking when we ask students to find

a solution to an equation, i.e., a value of the variable that makes the equation

true. In everyday language we think of a solution as a way to deal with a

difficult situation. We can contrast these typical and technical definitions

with sentences like the following.

• A solution to 2x + 3 = 5 is x = 1.

14.
170 Preparing Our Students to Read and Understand Mathematics

• A first step in a solution to finding a line that is parallel to y = 2x + 3

is to find the slope of the line.

Some other mathematical words that also could be confusing to students

because of a difference between the typical and technical meanings are open,

limit, union, ratio, acute, complementary, congruent, group, and function.

Use history and other context to help make sense of new vocabulary. If there

is a storyline, students may have an easier time remembering the word. Many

mathematical words are chosen for a reason; the words themselves are meant

to be a clue into the meaning. One idea from an interview is that before

introducing new words, you might deliberately design sentences that use

math jargon and ask students to guess at the meaning. By doing so, we are

getting students to reflect on why someone might choose the word and also

to activate prior knowledge related to the word. Once you have discussed

the technical definition of the word, you might then engage the class in a

discussion of what word they would choose if they were the mathematician

who got to name it. This technique can help to humanize mathematics. In

some cases, it is necessary to discuss how the word is related to another

language or why a particular language is used.

Although there are less techniques and research into teaching notation, math-

ematical notation is its own language, and learning this notation is similar

to learning new vocabulary. In this way, instruction on new notation should

be deliberate and meaningful. Emphasize to students why we are using no-

tation: to be more compact, to allow us to write more quickly, or to help

us remember something. Once again history can be a helpful tool in helping

students to see that someone selected this notation for some reason. Why?

What could we do differently? Is there a different notation that is better in

some ways? Notation may make things easier sometimes and harder other

times. The Frayer Model and other graphic organizers could also be used to

help students with notation. Model reading sentences containing the new no-

tation out loud while students follow along. Have students practice reading

the sentences out loud.

The following graphic organizer can help students with new notation and, by

forcing the students to write the meaning out in words, can help students to

see why the notation is useful. Here is a completed example of this model to

practice the notation for the complement of a set from basic set theory.

• A first step in a solution to finding a line that is parallel to y = 2x + 3

is to find the slope of the line.

Some other mathematical words that also could be confusing to students

because of a difference between the typical and technical meanings are open,

limit, union, ratio, acute, complementary, congruent, group, and function.

Use history and other context to help make sense of new vocabulary. If there

is a storyline, students may have an easier time remembering the word. Many

mathematical words are chosen for a reason; the words themselves are meant

to be a clue into the meaning. One idea from an interview is that before

introducing new words, you might deliberately design sentences that use

math jargon and ask students to guess at the meaning. By doing so, we are

getting students to reflect on why someone might choose the word and also

to activate prior knowledge related to the word. Once you have discussed

the technical definition of the word, you might then engage the class in a

discussion of what word they would choose if they were the mathematician

who got to name it. This technique can help to humanize mathematics. In

some cases, it is necessary to discuss how the word is related to another

language or why a particular language is used.

Although there are less techniques and research into teaching notation, math-

ematical notation is its own language, and learning this notation is similar

to learning new vocabulary. In this way, instruction on new notation should

be deliberate and meaningful. Emphasize to students why we are using no-

tation: to be more compact, to allow us to write more quickly, or to help

us remember something. Once again history can be a helpful tool in helping

students to see that someone selected this notation for some reason. Why?

What could we do differently? Is there a different notation that is better in

some ways? Notation may make things easier sometimes and harder other

times. The Frayer Model and other graphic organizers could also be used to

help students with notation. Model reading sentences containing the new no-

tation out loud while students follow along. Have students practice reading

the sentences out loud.

The following graphic organizer can help students with new notation and, by

forcing the students to write the meaning out in words, can help students to

see why the notation is useful. Here is a completed example of this model to

practice the notation for the complement of a set from basic set theory.

15.
Melanie Butler 171

Symbol Read out loud as (may be more

A0 than one way)

A complement

the complement of the set A

Example in symbols Example in words

U = {1, 2, 3, 4, 5} The universal set U contains the

A = {1, 2, 3} elements 1, 2, 3, 4, and 5. If A

A0 = {4, 5} is the set consisting of 1, 2, and

3, then A complement is the set

consisting of 4 and 5.

Alternative notations Other ways the same symbol is

AC used single tic mark is also used

for first derivative

3.2. Background Knowledge

Before students complete an assigned reading, instructors can aid comprehen-

sion of the reading by helping students activate their background knowledge

on the topic. In addition, taking time to think about background knowledge

helps motivate students to complete the reading and gives more of a pur-

pose to the assignment. Students might use the chapter or section titles, the

vocabulary or other notation, or a prompt provided by the instructor. Just

giving students time to brainstorm about background knowledge could be

helpful, but the following are some more formal strategies.

Give students time to answer the following questions and then discuss in

small groups or as a class.

1. Have you ever seen these words before? Was it in another class? Was

it in real life?

2. Do you know any words or concepts that are related to these words?

How might they be related?

3. Did you ever wonder about these words or ideas in the past? Were you

ever confused by something related to these words or ideas?

Have students brainstorm questions they might have on a topic. After read-

ing, have the class come back to this list of questions and decide if the

Symbol Read out loud as (may be more

A0 than one way)

A complement

the complement of the set A

Example in symbols Example in words

U = {1, 2, 3, 4, 5} The universal set U contains the

A = {1, 2, 3} elements 1, 2, 3, 4, and 5. If A

A0 = {4, 5} is the set consisting of 1, 2, and

3, then A complement is the set

consisting of 4 and 5.

Alternative notations Other ways the same symbol is

AC used single tic mark is also used

for first derivative

3.2. Background Knowledge

Before students complete an assigned reading, instructors can aid comprehen-

sion of the reading by helping students activate their background knowledge

on the topic. In addition, taking time to think about background knowledge

helps motivate students to complete the reading and gives more of a pur-

pose to the assignment. Students might use the chapter or section titles, the

vocabulary or other notation, or a prompt provided by the instructor. Just

giving students time to brainstorm about background knowledge could be

helpful, but the following are some more formal strategies.

Give students time to answer the following questions and then discuss in

small groups or as a class.

1. Have you ever seen these words before? Was it in another class? Was

it in real life?

2. Do you know any words or concepts that are related to these words?

How might they be related?

3. Did you ever wonder about these words or ideas in the past? Were you

ever confused by something related to these words or ideas?

Have students brainstorm questions they might have on a topic. After read-

ing, have the class come back to this list of questions and decide if the

16.
172 Preparing Our Students to Read and Understand Mathematics

1. Answered the question explicitly.

2. Did not answer the question. In this case, help the students brainstorm

about where they might find the answer to the question. Or maybe the

students can consider if the question is really related to the topic or

related to something else.

3. Assumed the students already knew the answer to the question. If so,

do the students still have the question? Where they able to infer an

answer to the question by information given in the reading?

The K-W-L strategy, developed by Donna Ogle [23], has the students make

a chart with three columns: Know, Want to Know, and Learned. The first

two columns are filled out as pre reading. The final column is filled out after

completing the reading.

3.3. Structure Analysis

In my interviews with faculty from other disciplines, instructors often talked

about the need to look for structure in a reading, such as looking for words

that divide or transition words. These same ideas can apply to looking for

structure in mathematics texts. As mathematicians, we are used to a certain

structure, which may be foreign to our students. Explicit instruction can help

students learn about the particulars of mathematics structure and syntax.

Students may need more explicit instruction in the meaning of the words

theorem, corollary, and lemma, for example. Faculty in other disciplines also

suggested rewriting each line of a reading in your own words or in shorthand.

This method leaves you with an outline of the reading, which can help you

look at the bigger picture and the overall structure.

3.4. Paced Reading

In many disciplines, mathematics included, students may not understand

that reading takes a long time. Other disciplines have developed techniques

to help students slow down when they are reading and to identify areas of

confusion. You might have students read the first paragraph and under-

line words they don’t know. Various techniques, such as writing all of the

unknown words on the board and going over them or having small group

discussions, can be used to help clarify points of confusion.

1. Answered the question explicitly.

2. Did not answer the question. In this case, help the students brainstorm

about where they might find the answer to the question. Or maybe the

students can consider if the question is really related to the topic or

related to something else.

3. Assumed the students already knew the answer to the question. If so,

do the students still have the question? Where they able to infer an

answer to the question by information given in the reading?

The K-W-L strategy, developed by Donna Ogle [23], has the students make

a chart with three columns: Know, Want to Know, and Learned. The first

two columns are filled out as pre reading. The final column is filled out after

completing the reading.

3.3. Structure Analysis

In my interviews with faculty from other disciplines, instructors often talked

about the need to look for structure in a reading, such as looking for words

that divide or transition words. These same ideas can apply to looking for

structure in mathematics texts. As mathematicians, we are used to a certain

structure, which may be foreign to our students. Explicit instruction can help

students learn about the particulars of mathematics structure and syntax.

Students may need more explicit instruction in the meaning of the words

theorem, corollary, and lemma, for example. Faculty in other disciplines also

suggested rewriting each line of a reading in your own words or in shorthand.

This method leaves you with an outline of the reading, which can help you

look at the bigger picture and the overall structure.

3.4. Paced Reading

In many disciplines, mathematics included, students may not understand

that reading takes a long time. Other disciplines have developed techniques

to help students slow down when they are reading and to identify areas of

confusion. You might have students read the first paragraph and under-

line words they don’t know. Various techniques, such as writing all of the

unknown words on the board and going over them or having small group

discussions, can be used to help clarify points of confusion.

17.
Melanie Butler 173

Another idea is to have students read out loud. Faculty that I interviewed

mentioned that this helps slow down thinking. In addition, faculty mentioned

that this technique has an equalizing effect on the class because everyone has

then had the same experience with a reading. Also, words have a different

impact when heard out loud. In mathematics, reading out loud can also help

students learn new notation as they are forced to make connections between

the notation and the meaning of the notation.

3.5. Anticipation Guides

Several sources suggest giving students Anticipation Guides before a reading

(for example, [1] and [26]). When using this technique, you start by giv-

ing students a set of true/false statements about the topic in the reading.

Researchers suggest using sources of common confusion to write statements.

In addition, you may use Bloom’s taxonomy [5], or another taxonomy, to

incorporate true/false statements at different levels of reasoning.

Before the reading students decide if they think the statements are true,

false, or sometimes true. Students should record their predictions. After

completing the reading, students go back to the same set of statements and

use the reading to justify if the statements are true, false, or sometimes

true. Whenever possible, students should use the text for justification by

writing down page numbers, theorem numbers, etc. Students should also

note the differences between their pre reading assessment and post reading

assessment. What changed? Why? It may be the case that not every answer

can be found explicitly in the reading. In this case, the instructor has an

opportunity to model how you reason out an answer from the information

provided in the reading.

4. Conclusion

The goal of this article is to motivate college math instructors to include

more instruction for reading and comprehending mathematics. In addition,

the article details activities for instructors to include in their classes to pre-

pare students to read on their own. A follow-up article is planned to detail

activities for students to help during reading and after reading.

Another idea is to have students read out loud. Faculty that I interviewed

mentioned that this helps slow down thinking. In addition, faculty mentioned

that this technique has an equalizing effect on the class because everyone has

then had the same experience with a reading. Also, words have a different

impact when heard out loud. In mathematics, reading out loud can also help

students learn new notation as they are forced to make connections between

the notation and the meaning of the notation.

3.5. Anticipation Guides

Several sources suggest giving students Anticipation Guides before a reading

(for example, [1] and [26]). When using this technique, you start by giv-

ing students a set of true/false statements about the topic in the reading.

Researchers suggest using sources of common confusion to write statements.

In addition, you may use Bloom’s taxonomy [5], or another taxonomy, to

incorporate true/false statements at different levels of reasoning.

Before the reading students decide if they think the statements are true,

false, or sometimes true. Students should record their predictions. After

completing the reading, students go back to the same set of statements and

use the reading to justify if the statements are true, false, or sometimes

true. Whenever possible, students should use the text for justification by

writing down page numbers, theorem numbers, etc. Students should also

note the differences between their pre reading assessment and post reading

assessment. What changed? Why? It may be the case that not every answer

can be found explicitly in the reading. In this case, the instructor has an

opportunity to model how you reason out an answer from the information

provided in the reading.

4. Conclusion

The goal of this article is to motivate college math instructors to include

more instruction for reading and comprehending mathematics. In addition,

the article details activities for instructors to include in their classes to pre-

pare students to read on their own. A follow-up article is planned to detail

activities for students to help during reading and after reading.

18.
174 Preparing Our Students to Read and Understand Mathematics

[1] Anne Adams, Jerine Pegg, and Melissa Case, “Anticipation guides:

Reading for mathematics understanding”, Mathematics Teacher, Vol-

ume 108 Issue 7 (March 2015), pages 498–504.

[2] Patricia Alexander, “The development of expertise: The journey from

acclimation to proficiency”, Educational Researcher, Volume 32 Number

8 (2003), pages 10–14.

[3] Patricia Alexander, “A Model of Domain Learning: Reinterpreting Ex-

pertise as a Multidimensional, Multistage Process”, in Motivation, Emo-

tion, and Cognition: Integrative Perspectives on Intellectual Functioning

and Development, edited by David Yun Dai and Robert J. Sternberg (L.

Erlbaum Associates, Mahwah, NJ, 2004), pages 273–298.

[4] Patricia Alexander and Emily Fox, “A historical perspective on reading

research and practice, redux”, in Theoretical Models and Processes of

Reading, 6th ed., edited by Donna Alvermann, Norman Unrau, and

Robert Ruddell, (International Reading Association, Newark DE, 2013),

pages 3–46.

[5] Benjamin Bloom, Max Englehart, Edward Furst, Walker Hill, and David

Krathwohl, Taxonomy of Educational Objectives: The Classification of

Educational Goals. Handbook I: Cognitive Domain, Longmans, Green,

and Company Limited, London, 1956.

[6] Richard Bullock and Richard Millman, “Mathematicians’ Concepts of

Audience in Mathematics Textbook Writing”, Problems, Resources, and

Issues in Mathematics Undergraduate Studies, Volume 2 Number 4

(1992), pages 335–347.

[7] Victor Byers, “Why study the history of mathematics?”, International

Journal of Mathematical Education in Science and Technology, Volume

13 Issue 1 (1982), pages 59–66.

[8] Olivia M. Carducci, “Card trick exercise leads to improved reading of

mathematics texts”, Problems, Resources, and Issues in Mathematics

Undergraduate Studies, (2018).

[1] Anne Adams, Jerine Pegg, and Melissa Case, “Anticipation guides:

Reading for mathematics understanding”, Mathematics Teacher, Vol-

ume 108 Issue 7 (March 2015), pages 498–504.

[2] Patricia Alexander, “The development of expertise: The journey from

acclimation to proficiency”, Educational Researcher, Volume 32 Number

8 (2003), pages 10–14.

[3] Patricia Alexander, “A Model of Domain Learning: Reinterpreting Ex-

pertise as a Multidimensional, Multistage Process”, in Motivation, Emo-

tion, and Cognition: Integrative Perspectives on Intellectual Functioning

and Development, edited by David Yun Dai and Robert J. Sternberg (L.

Erlbaum Associates, Mahwah, NJ, 2004), pages 273–298.

[4] Patricia Alexander and Emily Fox, “A historical perspective on reading

research and practice, redux”, in Theoretical Models and Processes of

Reading, 6th ed., edited by Donna Alvermann, Norman Unrau, and

Robert Ruddell, (International Reading Association, Newark DE, 2013),

pages 3–46.

[5] Benjamin Bloom, Max Englehart, Edward Furst, Walker Hill, and David

Krathwohl, Taxonomy of Educational Objectives: The Classification of

Educational Goals. Handbook I: Cognitive Domain, Longmans, Green,

and Company Limited, London, 1956.

[6] Richard Bullock and Richard Millman, “Mathematicians’ Concepts of

Audience in Mathematics Textbook Writing”, Problems, Resources, and

Issues in Mathematics Undergraduate Studies, Volume 2 Number 4

(1992), pages 335–347.

[7] Victor Byers, “Why study the history of mathematics?”, International

Journal of Mathematical Education in Science and Technology, Volume

13 Issue 1 (1982), pages 59–66.

[8] Olivia M. Carducci, “Card trick exercise leads to improved reading of

mathematics texts”, Problems, Resources, and Issues in Mathematics

Undergraduate Studies, (2018).

19.
Melanie Butler 175

[9] Dolores Durkin, “What classroom observations reveal about reading

comprehension instruction”, Reading Research Quarterly, Volume 14

Number 4 (1978), pages 481–533.

[10] Zhihui Fang and Suzanne Coatoam “Disciplinary literacy: What you

want to know about it”, Journal of Adolescent & Adult Literacy, Volume

56 Number 8 (May 2013), pages 627–632.

[11] Jeff D. Farmer and Jane F. Schielack, “Mathematics readings for non-

mathematics majors”, Problems, Resources, and Issues in Mathematics

Undergraduate Studies, Volume 2 Issue 4 (1992), pages 357–369.

[12] Dorothy Frayer, Wayne Frederick, and Herbert Klausmeier, “A Schema

for Testing the Level of Cognitive Mastery”, Working Paper No. 16

Wisconsin Research and Development Center, Madison: University of

Wisconsin, 1969.

[13] Judith V. Grabiner, “The mathematician, the historian, and the history

of mathematics”, Historia Mathematica, Volume 2 (1975), pages 439-

447.

[14] John Guthrie and Allan Wigfield, “Engagement and motivation in Read-

ing”, in Reading Research Handbook, Volume III, edited by M. L. Kamil,

P. B. Mosenthal, P. D. Pearson, and R. Barr (Lawrence Erlbaum Asso-

ciates, Mahwah NJ, 2000), pages 403–424.

[15] Gizem Karaali, “An ‘Unreasonable’ Component to a Reasonable Course:

Readings for a Transitional Class”, in Using the Philosophy of Mathe-

matics in Teaching Undergraduate Mathematics, edited by Bonnie Gold,

Carl Behrens, and Roger Simons (Mathematical Association of America,

Washington DC, 2017), pages 107–118.

[16] Dale Johnson and P. David Pearson, Teaching Reading Vocabulary, 2nd

ed., Holt, Rinehart and Winston, New York, 1984.

[17] Joan Kenney, Euthecia Hancewicz, Loretta Heuer, Diana Metsisto, and

Cynthia Tuttle, Literacy Strategies for Improving Mathematics Instruc-

tion, Association for Supervision and Curriculum Development, Alexan-

dria VA, 2005.

[9] Dolores Durkin, “What classroom observations reveal about reading

comprehension instruction”, Reading Research Quarterly, Volume 14

Number 4 (1978), pages 481–533.

[10] Zhihui Fang and Suzanne Coatoam “Disciplinary literacy: What you

want to know about it”, Journal of Adolescent & Adult Literacy, Volume

56 Number 8 (May 2013), pages 627–632.

[11] Jeff D. Farmer and Jane F. Schielack, “Mathematics readings for non-

mathematics majors”, Problems, Resources, and Issues in Mathematics

Undergraduate Studies, Volume 2 Issue 4 (1992), pages 357–369.

[12] Dorothy Frayer, Wayne Frederick, and Herbert Klausmeier, “A Schema

for Testing the Level of Cognitive Mastery”, Working Paper No. 16

Wisconsin Research and Development Center, Madison: University of

Wisconsin, 1969.

[13] Judith V. Grabiner, “The mathematician, the historian, and the history

of mathematics”, Historia Mathematica, Volume 2 (1975), pages 439-

447.

[14] John Guthrie and Allan Wigfield, “Engagement and motivation in Read-

ing”, in Reading Research Handbook, Volume III, edited by M. L. Kamil,

P. B. Mosenthal, P. D. Pearson, and R. Barr (Lawrence Erlbaum Asso-

ciates, Mahwah NJ, 2000), pages 403–424.

[15] Gizem Karaali, “An ‘Unreasonable’ Component to a Reasonable Course:

Readings for a Transitional Class”, in Using the Philosophy of Mathe-

matics in Teaching Undergraduate Mathematics, edited by Bonnie Gold,

Carl Behrens, and Roger Simons (Mathematical Association of America,

Washington DC, 2017), pages 107–118.

[16] Dale Johnson and P. David Pearson, Teaching Reading Vocabulary, 2nd

ed., Holt, Rinehart and Winston, New York, 1984.

[17] Joan Kenney, Euthecia Hancewicz, Loretta Heuer, Diana Metsisto, and

Cynthia Tuttle, Literacy Strategies for Improving Mathematics Instruc-

tion, Association for Supervision and Curriculum Development, Alexan-

dria VA, 2005.

20.
176 Preparing Our Students to Read and Understand Mathematics

[18] Jonna Kulikowich and Patricia Alexander, “Intentionality to learn in

an academic domain”, Early Education and Development, Volume 21

Number 5 (2010), pages 724–743.

[19] Liliana Maggioni, Bruce VanSledright, and Patricia Alexander, “Walk-

ing on the borders: A measure of epistemic cognition in history”, The

Journal of Experimental Education, Volume 77 Number 3 (2009), pages

187–214.

[20] Margaret McKeown, Isabel Beck, and Ronette Blake, “Rethinking read-

ing comprehension instruction: A comparison of instruction for strate-

gies and content approaches”, Reading Research Quarterly, Volume 44

Number 3 (2009), pages 218–253.

[21] P. Karen Murphy, Ian Wilkinson, Anna Soter, Maeghan Hennessey, and

John Alexander, “Examining the effects of classroom discussion on stu-

dents’ comprehension of text: A meta-analysis”, Journal of Educational

Psychology, Volume 101 Number 3 (2009), pages 740–764.

[22] National Council of Teachers of Mathematics (NCTM), Principles and

Standards for School Mathematics, NCTM, Reston VA, 2000.

[23] Donna Ogle, “K-W-L: A teaching model that develops active reading of

expository text”, The Reading Teacher, Volume 39 Number 6 (1986),

pages 564–570.

[24] P. D. Pearson and Dale Johnson, Teaching Reading Comprehension,

Holt, Rinehart & Winston, New York, 1978.

[25] Karen Saxe and Linda Braddy, A Common Vision for Mathemati-

cal Sciences Programs in 2025, Mathematical Association of Amer-

ica, Washington DC, 2015; available at http://www.maa.org/sites/

default/files/pdf/CommonVisionFinal.pdf, last accessed on Jan-

uary 29, 2019.

[26] Timothy Shanahan and Cynthia Shanahan, “What is disciplinary liter-

acy and why does it matter?”, Topics in Language Disorders, Volume

32 Number 1 (2012), pages 7–18.

[27] S. E. Wade and E. B. Moje, “The role of text in classroom learning”,

in The Handbook of Research on Reading Volume III, edited by M.L.

[18] Jonna Kulikowich and Patricia Alexander, “Intentionality to learn in

an academic domain”, Early Education and Development, Volume 21

Number 5 (2010), pages 724–743.

[19] Liliana Maggioni, Bruce VanSledright, and Patricia Alexander, “Walk-

ing on the borders: A measure of epistemic cognition in history”, The

Journal of Experimental Education, Volume 77 Number 3 (2009), pages

187–214.

[20] Margaret McKeown, Isabel Beck, and Ronette Blake, “Rethinking read-

ing comprehension instruction: A comparison of instruction for strate-

gies and content approaches”, Reading Research Quarterly, Volume 44

Number 3 (2009), pages 218–253.

[21] P. Karen Murphy, Ian Wilkinson, Anna Soter, Maeghan Hennessey, and

John Alexander, “Examining the effects of classroom discussion on stu-

dents’ comprehension of text: A meta-analysis”, Journal of Educational

Psychology, Volume 101 Number 3 (2009), pages 740–764.

[22] National Council of Teachers of Mathematics (NCTM), Principles and

Standards for School Mathematics, NCTM, Reston VA, 2000.

[23] Donna Ogle, “K-W-L: A teaching model that develops active reading of

expository text”, The Reading Teacher, Volume 39 Number 6 (1986),

pages 564–570.

[24] P. D. Pearson and Dale Johnson, Teaching Reading Comprehension,

Holt, Rinehart & Winston, New York, 1978.

[25] Karen Saxe and Linda Braddy, A Common Vision for Mathemati-

cal Sciences Programs in 2025, Mathematical Association of Amer-

ica, Washington DC, 2015; available at http://www.maa.org/sites/

default/files/pdf/CommonVisionFinal.pdf, last accessed on Jan-

uary 29, 2019.

[26] Timothy Shanahan and Cynthia Shanahan, “What is disciplinary liter-

acy and why does it matter?”, Topics in Language Disorders, Volume

32 Number 1 (2012), pages 7–18.

[27] S. E. Wade and E. B. Moje, “The role of text in classroom learning”,

in The Handbook of Research on Reading Volume III, edited by M.L.

21.
Melanie Butler 177

Kamil, P.B. Mosenthal, P. D. Pearson, and R. Barr (Lawrence Erlbaum

Associates, Mahwah NJ, 2000), pages 609–627.

[28] Keith Weber, “Effective proof reading strategies for comprehending

mathematical proofs”, International Journal of Research in Undergrad-

uate Mathematics Education, Volume 1 Issue 3 (2015), pages 289–314.

[29] Aaron Weinberg, Emilie Wiesner, Bret Benesh and Timothy Boester,

“Undergraduate students’ self-reported use of mathematics textbooks”,

Problems, Resources, and Issues in Mathematics Undergraduate Studies,

Volume 22 Issue 2 (2012), pages 152–175.

[30] University of Cambridge Mathematics Faculty Mathematical Read-

ing List, March 20, 2015, available at https://www.maths.cam.

ac.uk/sites/www.maths.cam.ac.uk/files/pre2014/undergrad/

admissions/readinglist.pdf, last accessed on January 28, 2019.

Kamil, P.B. Mosenthal, P. D. Pearson, and R. Barr (Lawrence Erlbaum

Associates, Mahwah NJ, 2000), pages 609–627.

[28] Keith Weber, “Effective proof reading strategies for comprehending

mathematical proofs”, International Journal of Research in Undergrad-

uate Mathematics Education, Volume 1 Issue 3 (2015), pages 289–314.

[29] Aaron Weinberg, Emilie Wiesner, Bret Benesh and Timothy Boester,

“Undergraduate students’ self-reported use of mathematics textbooks”,

Problems, Resources, and Issues in Mathematics Undergraduate Studies,

Volume 22 Issue 2 (2012), pages 152–175.

[30] University of Cambridge Mathematics Faculty Mathematical Read-

ing List, March 20, 2015, available at https://www.maths.cam.

ac.uk/sites/www.maths.cam.ac.uk/files/pre2014/undergrad/

admissions/readinglist.pdf, last accessed on January 28, 2019.