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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.
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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
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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 .
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2.
Preparing Our Students to Read and Understand
Mathematics
Melanie Butler
Department of Mathematics & Computer Science, Mount St. Mary’s University,
Maryland, USA
[email protected]
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)
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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.
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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.”
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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”.
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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:
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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.
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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.
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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.
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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.
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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;
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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
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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.
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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.
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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
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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.
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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.
18.
174 Preparing Our Students to Read and Understand Mathematics
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