Beginning mathematical writing
assignments
August 6, 2019†
Abstract: Writing assignments in any mathematics course always present
several challenges, particularly in lower-level classes where the students are
not expecting to write more than a few words at a time. Developed based on
strategies from several sources, the two small writing assignments included in
this paper represent a gentle introduction to the writing of mathematics and
can be utilized in a variety of low-to-middle level courses in a mathematics
major.
Introduction
Many math majors mistakenly believe that they will not be expected to write as part
of their coursework. Many algebra, precalculus, and calculus classes require little to no
mathematical writing, even though opportunities abound in problem-solving, exposition,
or even self-reflection after an assignment. Students, in turn, become accustomed to this
lack of writing and fail to see the logical nature of mathematics and ultimately struggle
with expressing themselves through writing.
The short writing assignments presented in this article represent our attempt to solve
several problems at once. First, these assignments provide a guided step into the formal
writing of mathematics. Second, they were used as part of a larger research project
comparing the progress of students from two different classes at two different universities:
one class with an intervention of a larger writing assignment (see Magnant, Nasseh, &
Flateby, 2016) and the other without. We omit the details of this larger assignment as
our focus here is on the smaller writing assignments. Finally, these assignments, with
minimal background introduction, can be assigned in virtually any course within a math
∗Salisbury University, Clayton State University, Georgia Southern University, adhalperin@salisbury.edu,
dr.colton.magnant@gmail.com, zmagnant@georgiasouthern.edu. Copyright 2019 Alexander Halperin,
Colton Magnant, and Zhoujun Magnant. This work is licensed under a Creative Commons AttributionNonCommercial
4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/).
†Submitted, 7/18/2017; Accepted, 4/27/2019.
program. In short, they were developed as both a measurement tool and an introduction
to formal mathematical writing.Students wrote one proof for each assignment. In the first writing assignment, students
determined whether a cat-and-mouse chase could pass exactly once through every open
door and window in a house, whose floor plan was given. The second assignment, “Poker
Hands,” required students to determine the winner of a three-player poker game by
ranking three poker hands from least to most probable. For each of these assignments,
students submitted their proof along with an abstract, an introduction, their results, and
a conclusion.
The writing assignments were developed using a combination of strategies and ideas from
Walvoord (2014), Bahls (2012), Bean (2011), and Crannell et al. (2004) as well as ideas
stemming from personal experience taking and teaching courses in mathematics. The
assignments foster effective writing habits and, at the same time, develop students’ skills
in the areas of argumentation, analysis, and synthesis. In order to prove their claimed
solution, students must argue using credible evidence and supporting logic. Effective
analysis and description of the situation in both assignments provides the framework
necessary for a successful complete solution. Finally, an element of synthesis is expected
in the summarizing conclusion where students must consider a possible “natural” next
step as a direction for future work.
Both writing assignments were assigned in each of two different classes, one at each of
the two different universities. Salisbury University (SU) is a mid-sized public regional
comprehensive university, while Georgia Southern University (GSU) is a large public
regional comprehensive university. At SU, the assignments were used in an introductory
discrete mathematics course. At GSU, the assignments were used in a course on introduction
to proofs called Mathematical Structures. Each class had about 30 students,
primarily second-year undergraduates with a handful of first-year and third-year students
as well. In a survey of the students in the class, most said they had never written more
than a sentence or two in a math class. Both classes were learning Claim-Proof form1 of
mathematical writing so an additional goal of these assignments was to reinforce this
writing style.
In the rest of this essay, we describe how we used these assignments to teach students
effective writing in mathematics. We begin with a discussion of the in-class group activities
before describing our findings when we allowed the students to work independently. Finally,
we offer some suggestions for what these findings mean for the teaching of mathematics
and mathematical writing.
Whole Class Practice
For each of the writing assignments, students spent a day in class solving a relatedproblem followed by a day devoted to formulating their ideas into paragraphs. In
doing so, they mimicked the standard mathematical problem-solving process:
discover the solution and then formally write it up. The mathematical content of
the assignments was deliberately involved—students needed to spend the entire period
discussing, understanding and answering the problem—but not terribly difficult so that
students could focus on the writing after class. The class was designed to be an interactive
experience: students worked on the problem in groups of three to four. The instructor
toured the classroom to sort out misconceptions and to ensure that all group members
were engaged. With around fifteen minutes left in the period, the instructors halted
group work to have students lead a discussion about the solution along with their
problem-solving process.
The next day, we moved away from problem-solving and began our focus on writing
by first handing out the assignment, which required a formal essay presentation of
their solution(s). Neither assignment was long, requiring only two to three pages. The
primary focus of both assignments was on the process of the writing and how to explain
solutions clearly, precisely, and concisely. We required that each paper contain the four
semi-standard sections found in most mathematical papers: an abstract, introduction,
main results, and conclusion. The assignment document contained brief summaries of
what type of content was expected in each section, and we provided ongoing classroom
discussion, guidance, and clarification throughout the process, including our reasoning
for requiring the four-section format. Armed with at least the bulk of a solution to
the mathematical content of the questions and, in most cases, a rough outline of their
solutions, the students proceeded home to complete the papers. Anecdotally, students
suggested that they were better able to synthesize information through the writing than
standard exercises, and they appreciated the experience. Students were expected to
complete each assignment in about a week.
In both classes at the two different universities, the student products were scored using
the same writing rubric (see the first appendix for a sample from the rubric, 3 of the 17
traits) for consistency. This rubric was developed primarily based on the well-thought-out
Georgia Southern University Quality Enhancement Plan writing rubric (2015) but also
drew on insights from Bahls (2012). This rubric was used since it provides a reliable and
consistent measure of the different components of the written products that we wanted
to measure for this project.
work and assigned them the formal writing problem. Recall that the first assignment
involves a cat chasing a mouse around a house, and the question involves deciding (and
proving) whether or not the pair can pass through all of the openings (doors and windows)
in the house without repeating any. In the second question, a heated debate over a poker
match requires a determination of which hand is least likely to appear and, therefore,
more valuable.
The playful nature2 of both questions was intended to welcome, rather than intimidate,
the assignments was deliberately involved—students needed to spend the entire period
discussing, understanding and answering the problem—but not terribly difficult so that
students could focus on the writing after class. The class was designed to be an interactive
experience: students worked on the problem in groups of three to four. The instructor
toured the classroom to sort out misconceptions and to ensure that all group members
were engaged. With around fifteen minutes left in the period, the instructors halted
group work to have students lead a discussion about the solution along with their
problem-solving process.
The next day, we moved away from problem-solving and began our focus on writing
by first handing out the assignment, which required a formal essay presentation of
their solution(s). Neither assignment was long, requiring only two to three pages. The
primary focus of both assignments was on the process of the writing and how to explain
solutions clearly, precisely, and concisely. We required that each paper contain the four
semi-standard sections found in most mathematical papers: an abstract, introduction,
main results, and conclusion. The assignment document contained brief summaries of
what type of content was expected in each section, and we provided ongoing classroom
discussion, guidance, and clarification throughout the process, including our reasoning
for requiring the four-section format. Armed with at least the bulk of a solution to
the mathematical content of the questions and, in most cases, a rough outline of their
solutions, the students proceeded home to complete the papers. Anecdotally, students
suggested that they were better able to synthesize information through the writing than
standard exercises, and they appreciated the experience. Students were expected to
complete each assignment in about a week.
In both classes at the two different universities, the student products were scored using
the same writing rubric (see the first appendix for a sample from the rubric, 3 of the 17
traits) for consistency. This rubric was developed primarily based on the well-thought-out
Georgia Southern University Quality Enhancement Plan writing rubric (2015) but also
drew on insights from Bahls (2012). This rubric was used since it provides a reliable and
consistent measure of the different components of the written products that we wanted
to measure for this project.
Trying It on Their Own
For each assignment, after practicing together as a class, we moved toward independentwork and assigned them the formal writing problem. Recall that the first assignment
involves a cat chasing a mouse around a house, and the question involves deciding (and
proving) whether or not the pair can pass through all of the openings (doors and windows)
in the house without repeating any. In the second question, a heated debate over a poker
match requires a determination of which hand is least likely to appear and, therefore,
more valuable.
The playful nature2 of both questions was intended to welcome, rather than intimidate,
students when they first read the assignment in class. Immediately after reading the
problem, students received a handout with questions pertaining to the paper. For the
rest of the class period, students worked in groups to discover a solution. The instructor
toured the classroom, answering minor questions about the mathematics when necessary,
and then led a discussion at the end regarding the solution. We detail this approach for
each assignment in the following sections. In subsequent class meetings, the instructors
reminded the students of due dates, and correspondingly where they should be in the
process of the writing. Relevant discussions of revision and editing were also included in
classroom discussions.
Students collaborated during class but were expected to each write their own separate
paper, as opposed to Latulippe and Latulippe’s assignments (2014), in which three to
four student groups turned in a single essay. This was to ensure that each student
was responsible for their own writing, as we prioritized writing over problem-solving.
Although the QEP rubric weights math and writing equally, most students understood
the solutions to both writing assignments by the end of the lecture, meaning that their
scores for the mathematical part should have been high with little variance.
Cat and Mouse
For the “Cat and Mouse” assignment, students first explored the famous “Seven Bridges
of Königsberg problem” (Paoletti, 2011), in which a traveler tries, in a continuous route,
to cross each of seven bridges in Königsberg, Prussia exactly once (see supplementary
materials for this in-class assignment). After some trial and error, students discovered
that such a route is impossible, along with the realization that the lack of solution must
be proved, rather than asserted. Listed below the problem statement were the steps to
the proof, out of order3
, for students to rearrange. At the end of the exercise, the class
recapped the argument to the instructor in their own words. The instructor then asked
students to partition the solution into paragraphs to best organize the ideas presented in
the proof. From there, students had the necessary mathematical tools to attempt the
first writing assignment. As a matter of fact, the “Cat and Mouse” problem has the same
solution as the Seven Bridges problem: the fact that there exist more than two rooms
with an odd number of doorways plus windows means the cat and mouse cannot pass
through every open door and window exactly once.
proper motivation and background. The following day, we demonstrated examples of
complete written solutions to similar problems and discussed advantages and disadvantages
of each along with recommendations for improvement. By writing mostly about
the framing of a problem, students were meant to see that a mathematical paper requires
far more than a problem statement and solution. Additionally, in following the example
solutions discussed in class, they were forced to write in paragraph form, a first for most
problem, students received a handout with questions pertaining to the paper. For the
rest of the class period, students worked in groups to discover a solution. The instructor
toured the classroom, answering minor questions about the mathematics when necessary,
and then led a discussion at the end regarding the solution. We detail this approach for
each assignment in the following sections. In subsequent class meetings, the instructors
reminded the students of due dates, and correspondingly where they should be in the
process of the writing. Relevant discussions of revision and editing were also included in
classroom discussions.
Students collaborated during class but were expected to each write their own separate
paper, as opposed to Latulippe and Latulippe’s assignments (2014), in which three to
four student groups turned in a single essay. This was to ensure that each student
was responsible for their own writing, as we prioritized writing over problem-solving.
Although the QEP rubric weights math and writing equally, most students understood
the solutions to both writing assignments by the end of the lecture, meaning that their
scores for the mathematical part should have been high with little variance.
Cat and Mouse
For the “Cat and Mouse” assignment, students first explored the famous “Seven Bridges
of Königsberg problem” (Paoletti, 2011), in which a traveler tries, in a continuous route,
to cross each of seven bridges in Königsberg, Prussia exactly once (see supplementary
materials for this in-class assignment). After some trial and error, students discovered
that such a route is impossible, along with the realization that the lack of solution must
be proved, rather than asserted. Listed below the problem statement were the steps to
the proof, out of order3
, for students to rearrange. At the end of the exercise, the class
recapped the argument to the instructor in their own words. The instructor then asked
students to partition the solution into paragraphs to best organize the ideas presented in
the proof. From there, students had the necessary mathematical tools to attempt the
first writing assignment. As a matter of fact, the “Cat and Mouse” problem has the same
solution as the Seven Bridges problem: the fact that there exist more than two rooms
with an odd number of doorways plus windows means the cat and mouse cannot pass
through every open door and window exactly once.
Given that the students could now easily solve the “Cat and Mouse” problem, the main
challenge of this assignment was for students to present a mathematical solution with itsproper motivation and background. The following day, we demonstrated examples of
complete written solutions to similar problems and discussed advantages and disadvantages
of each along with recommendations for improvement. By writing mostly about
the framing of a problem, students were meant to see that a mathematical paper requires
far more than a problem statement and solution. Additionally, in following the example
solutions discussed in class, they were forced to write in paragraph form, a first for most
According to the rubric scores, students particularly struggled with audience awareness
by failing to provide the necessary context and definitions needed for full understanding.
Although the target audience was not explicitly stated in the assignment prompt, the
students were instructed to write as if to an audience of their peers. This audience was
purposely chosen to encourage peer review in the revision process. Broadly speaking,
other areas of the rubric were in the acceptable range, especially the proof write-up. This
was not surprising since the question is simple to understand, and the students already
have a correct proof from their class notes.
Poker Hands
The purpose of the poker writing assignment was to again show students that mathematics
is primarily carried out in words, rather than symbols. Further, we wanted students to
understand the power of the “combinatorial proof,” in which quantities are counted using
basic multiplication, factorials, and combinations. A problem that could take hundreds of
algebraic calculations by brute force can sometimes be answered in a few short sentences
in a combinatorial proof, hence making it the far more desirable option.
The writing assignment on poker hands came at the end of a week’s worth of combinatorics
lessons. Students had studied the combination “n choose r,” or (
n
r
), and learned its
combinatorial definition (“(
n
r
) is the number of ways to choose a subset with r elements
out of a set with n elements”) as well as derived its algebraic formula
(
n
r
) = n!
(n−r)!r!
.
Further, they had dealt with counting problems, including a poker hand example and
several more involving playing cards. Most importantly, students had learned to solve
these problems by viewing (
n
r
) through its combinatorial definition, which emphasizes
exposition and conceptual understanding over calculation. Thus, they understood that
the algebraic formula is more a technical result and is only used at the end of a problem
to find an exact numerical answer.
At the beginning of the first class, we asked students to work in groups to count the
number of each type of poker hand (see the writing assignment, including the crossedout
hands). The professor roamed the classroom, sorting out any misconceptions and
correcting what are usually small errors. By the end of class, all student groups had
counted all or nearly all poker hands.
The main task for the writing assignment was for students to formally write their
classroom discoveries as a logical sequence of steps, in which they reformulate a poker
hand into the values and/or suits that were chosen to form it. Of course, they still
needed to provide background for this problem, but that was usually simpler than the
previous writing assignment. Most found the poker hand situation more gripping than
the cat-mouse chase, and many were eager to research the history of poker (and, in some
cases, discuss it at great length). In fact, some students took such interest with the
history of poker that it dominated the Introduction. To remedy this and keep the fo
by failing to provide the necessary context and definitions needed for full understanding.
Although the target audience was not explicitly stated in the assignment prompt, the
students were instructed to write as if to an audience of their peers. This audience was
purposely chosen to encourage peer review in the revision process. Broadly speaking,
other areas of the rubric were in the acceptable range, especially the proof write-up. This
was not surprising since the question is simple to understand, and the students already
have a correct proof from their class notes.
Poker Hands
The purpose of the poker writing assignment was to again show students that mathematics
is primarily carried out in words, rather than symbols. Further, we wanted students to
understand the power of the “combinatorial proof,” in which quantities are counted using
basic multiplication, factorials, and combinations. A problem that could take hundreds of
algebraic calculations by brute force can sometimes be answered in a few short sentences
in a combinatorial proof, hence making it the far more desirable option.
The writing assignment on poker hands came at the end of a week’s worth of combinatorics
lessons. Students had studied the combination “n choose r,” or (
n
r
), and learned its
combinatorial definition (“(
n
r
) is the number of ways to choose a subset with r elements
out of a set with n elements”) as well as derived its algebraic formula
(
n
r
) = n!
(n−r)!r!
.
Further, they had dealt with counting problems, including a poker hand example and
several more involving playing cards. Most importantly, students had learned to solve
these problems by viewing (
n
r
) through its combinatorial definition, which emphasizes
exposition and conceptual understanding over calculation. Thus, they understood that
the algebraic formula is more a technical result and is only used at the end of a problem
to find an exact numerical answer.
At the beginning of the first class, we asked students to work in groups to count the
number of each type of poker hand (see the writing assignment, including the crossedout
hands). The professor roamed the classroom, sorting out any misconceptions and
correcting what are usually small errors. By the end of class, all student groups had
counted all or nearly all poker hands.
The main task for the writing assignment was for students to formally write their
classroom discoveries as a logical sequence of steps, in which they reformulate a poker
hand into the values and/or suits that were chosen to form it. Of course, they still
needed to provide background for this problem, but that was usually simpler than the
previous writing assignment. Most found the poker hand situation more gripping than
the cat-mouse chase, and many were eager to research the history of poker (and, in some
cases, discuss it at great length). In fact, some students took such interest with the
history of poker that it dominated the Introduction. To remedy this and keep the fo
on the problem at hand, future assignments will specify a limit to the “history” portion
of the Introduction, and the rubric will be adjusted accordingly.
This being the second assignment, the instructors were able to discuss the issues with
audience awareness in the previous assignment by providing and discussing specific
examples in class so the scores in this area were slightly improved over the “Cat and
Mouse” assignment. Other areas where the students showed a bit of weakness was
paragraph structure, transition between paragraphs, and overall flow. In particular,
several students listed calculations for the three chosen hands with almost no discussion
in between. One way to combat this misunderstanding may be to create a different rubric
and provide it to students with the assignment. We discuss this at the end of the next
section and give such a rubric in the supplementary materials.
General Conclusion and Future Plans
Our modest goal was to use these writing assignments as an introduction to mathematical
writing, opening the door to the world of written mathematics. That said, given the ease
in which students were able to state their solutions, we believe we should add a degree
of difficulty to each assignment. For the “Cat and Mouse” assignment, this could mean
asking for a more general proof that any multigraph with more than two vertices of odd
degree has no Eulerian trail. For the poker assignment, we are considering requiring
students to find the probability of a “high card,” (the most difficult to explain), including
a wild card (i.e., a card that can take on any value or suit), or perhaps a variant game
with a different number of card values and/or suits.4
In the future, we hope to further incorporate a small reflection piece after each assignment
in which the students will reflect on their process of writing, which may (should) include
revision, peer-review, editing, etc. We hope to use this information to enlighten students
to the effective writing processes that not only make them stronger writers but also,
more immediately, result in better scores. Although explicit mention of process writing
was not included in the assignment prompts, it was discussed in class, particularly when
discussing outlines, revision, and peer review. The students outlined their solutions in
class and were encouraged to review and revise each other’s writing outside of class
since these were intended to be shorter writing assignments. We intend to include more
deliberate details about process writing in future iterations.
Broadly speaking, students show better performance when the audience for the assignment
is clear, supporting the theory in the MAA Instructional Practices Guide (2018). We
were pleasantly surprised at the amount and effectiveness of the peer reviews and intend
to be more intentional about the audience in further iterations of these assignments, in
particular, including explicit statements about audience in the assignment prompts.
We also plan to develop more advanced and challenging problems to further the students’
writing experience later in their mathematical careers. In both of these assignments,
although students spent a day on each of the “Seven Bridges of Königsberg”
of the Introduction, and the rubric will be adjusted accordingly.
This being the second assignment, the instructors were able to discuss the issues with
audience awareness in the previous assignment by providing and discussing specific
examples in class so the scores in this area were slightly improved over the “Cat and
Mouse” assignment. Other areas where the students showed a bit of weakness was
paragraph structure, transition between paragraphs, and overall flow. In particular,
several students listed calculations for the three chosen hands with almost no discussion
in between. One way to combat this misunderstanding may be to create a different rubric
and provide it to students with the assignment. We discuss this at the end of the next
section and give such a rubric in the supplementary materials.
General Conclusion and Future Plans
Our modest goal was to use these writing assignments as an introduction to mathematical
writing, opening the door to the world of written mathematics. That said, given the ease
in which students were able to state their solutions, we believe we should add a degree
of difficulty to each assignment. For the “Cat and Mouse” assignment, this could mean
asking for a more general proof that any multigraph with more than two vertices of odd
degree has no Eulerian trail. For the poker assignment, we are considering requiring
students to find the probability of a “high card,” (the most difficult to explain), including
a wild card (i.e., a card that can take on any value or suit), or perhaps a variant game
with a different number of card values and/or suits.4
In the future, we hope to further incorporate a small reflection piece after each assignment
in which the students will reflect on their process of writing, which may (should) include
revision, peer-review, editing, etc. We hope to use this information to enlighten students
to the effective writing processes that not only make them stronger writers but also,
more immediately, result in better scores. Although explicit mention of process writing
was not included in the assignment prompts, it was discussed in class, particularly when
discussing outlines, revision, and peer review. The students outlined their solutions in
class and were encouraged to review and revise each other’s writing outside of class
since these were intended to be shorter writing assignments. We intend to include more
deliberate details about process writing in future iterations.
Broadly speaking, students show better performance when the audience for the assignment
is clear, supporting the theory in the MAA Instructional Practices Guide (2018). We
were pleasantly surprised at the amount and effectiveness of the peer reviews and intend
to be more intentional about the audience in further iterations of these assignments, in
particular, including explicit statements about audience in the assignment prompts.
We also plan to develop more advanced and challenging problems to further the students’
writing experience later in their mathematical careers. In both of these assignments,
although students spent a day on each of the “Seven Bridges of Königsberg”
hands, the actual mathematical solutions were short, no longer than a paragraph each.
Although our students appreciated the succinctness of a well-written mathematical proof,
we do want them to experience writing a proof that, by itself, spans at least a page or,
better still, requires steps or lemmas to prove.
One further change, particularly for the poker assignment, could be to require students
to research a real-world combinatorial question. While similar to the approach used by
Pinter (Pinter, 2014), we would require students to find a question outside of class instead
of citing previous course material. Since, unlike Pinter, our classes contain exclusively
math majors and minors, we believe the added component of researching combinatorial
problems outside of class would be a fair, albeit challenging, additional requirement. Such
an assignment would require group work for both discovering and solving the problem. To
ensure that students choose an appropriately difficult problem, we would have students
first hand in their problems before writing the essay.
We have also considered changing the rubric to something shorter, more readable, and with
specific guidelines (e.g., “Utilize
52
5
in each proof”). We believe this may reduce student
forgetfulness or misunderstanding of assignment requirements, such as including ClaimProof
form for all five poker hand results or making conjectures in the conclusion. We
have included an outline of such a rubric for the poker hands assignment in supplementary
materials.
Assignment:
See the Supplementary Files for this article at thepromptjournal.com for a PDF facsimile
of the original formatting of this assignment.
Writing Rubric Excerpt
Does not
meet (1) Attempted (2)
Approaches
(3) Meets (4)
Exceeds
(5)
Assignment
Requirements
The
writer is
off topic
and/or
omits
most or
all of the
assignment
requirements.
The writer
addresses the
appropriate
topic but only
superficially
addresses the
assignment
requirements.
The writer
addresses
the
appropriate
topic and
meets the
assignment
requirements.
The writer
addresses the
appropriate
topic and
clearly and
correctly
fulfills each
aspect of the
assignment
requirements.
The writer
addresses
the appropriate
topic
and clearly,
correctly,
and
concisely
fulfills each
aspect of
the
assignment
requirement
Although our students appreciated the succinctness of a well-written mathematical proof,
we do want them to experience writing a proof that, by itself, spans at least a page or,
better still, requires steps or lemmas to prove.
One further change, particularly for the poker assignment, could be to require students
to research a real-world combinatorial question. While similar to the approach used by
Pinter (Pinter, 2014), we would require students to find a question outside of class instead
of citing previous course material. Since, unlike Pinter, our classes contain exclusively
math majors and minors, we believe the added component of researching combinatorial
problems outside of class would be a fair, albeit challenging, additional requirement. Such
an assignment would require group work for both discovering and solving the problem. To
ensure that students choose an appropriately difficult problem, we would have students
first hand in their problems before writing the essay.
We have also considered changing the rubric to something shorter, more readable, and with
specific guidelines (e.g., “Utilize
52
5
in each proof”). We believe this may reduce student
forgetfulness or misunderstanding of assignment requirements, such as including ClaimProof
form for all five poker hand results or making conjectures in the conclusion. We
have included an outline of such a rubric for the poker hands assignment in supplementary
materials.
Assignment:
See the Supplementary Files for this article at thepromptjournal.com for a PDF facsimile
of the original formatting of this assignment.
Writing Rubric Excerpt
Does not
meet (1) Attempted (2)
Approaches
(3) Meets (4)
Exceeds
(5)
Assignment
Requirements
The
writer is
off topic
and/or
omits
most or
all of the
assignment
requirements.
The writer
addresses the
appropriate
topic but only
superficially
addresses the
assignment
requirements.
The writer
addresses
the
appropriate
topic and
meets the
assignment
requirements.
The writer
addresses the
appropriate
topic and
clearly and
correctly
fulfills each
aspect of the
assignment
requirements.
The writer
addresses
the appropriate
topic
and clearly,
correctly,
and
concisely
fulfills each
aspect of
the
assignment
requirement
Prompt 3.2 2019
Does not
meet (1) Attempted (2)
Approaches
(3) Meets (4)
Exceeds
(5)
Reasoning
(Proof)
The
logical
connection
of
the argument
is
weak,
leaving
the argument
or
explanation
unclear.
A “proof
by example”
falls
here.
The reasoning
offers apparent
support for the
argument, but
the argument
or explanation
is weak.
Collectively,
the logic
offers
adequate
support for
the
argument,
but the
argument or
explanation
remains
unclear or
incomplete.
Collectively,
the logic
supports and
advances the
argument or
explanation
of the proof.
Collectively,
the logical
steps offer
compelling
support
which
clearly
advances
the
argument
or explanation
of the
proof.
Quality of
Details
Details
are superficial
or do
not
develop
the
proof.
Details are
loosely related
to the proof.
Many do not
provide
supporting
statements,
credible
evidence, or the
examples
necessary to
explain or
persuade
adequately.
Details are
related to
the proof
but
inconsistently
provide
supporting
statements,
credible
evidence, or
the
examples
necessary to
explain or
persuade
adequately.
Details
provide
supporting
statements,
credible
evidence, or
the examples
necessary to
explain or
persuade
adequately.
Compelling
details
provide
supporting
statements,
credible
evidence,
or the
examples
necessary
to explain
or persuade
effectively.
Assignment 1: Cat and Mouse
A cat chases a mouse in and out of a house. Due to the hot weather and malfunctioning
air conditioner, all doors and windows are open. This provides a rousing game of tag, as
both the mouse and the cat are small enough to fit through all doorways and window
frames. Is it possible for the cat and mouse to run through every doorway and wind
Does not
meet (1) Attempted (2)
Approaches
(3) Meets (4)
Exceeds
(5)
Reasoning
(Proof)
The
logical
connection
of
the argument
is
weak,
leaving
the argument
or
explanation
unclear.
A “proof
by example”
falls
here.
The reasoning
offers apparent
support for the
argument, but
the argument
or explanation
is weak.
Collectively,
the logic
offers
adequate
support for
the
argument,
but the
argument or
explanation
remains
unclear or
incomplete.
Collectively,
the logic
supports and
advances the
argument or
explanation
of the proof.
Collectively,
the logical
steps offer
compelling
support
which
clearly
advances
the
argument
or explanation
of the
proof.
Quality of
Details
Details
are superficial
or do
not
develop
the
proof.
Details are
loosely related
to the proof.
Many do not
provide
supporting
statements,
credible
evidence, or the
examples
necessary to
explain or
persuade
adequately.
Details are
related to
the proof
but
inconsistently
provide
supporting
statements,
credible
evidence, or
the
examples
necessary to
explain or
persuade
adequately.
Details
provide
supporting
statements,
credible
evidence, or
the examples
necessary to
explain or
persuade
adequately.
Compelling
details
provide
supporting
statements,
credible
evidence,
or the
examples
necessary
to explain
or persuade
effectively.
Assignment 1: Cat and Mouse
A cat chases a mouse in and out of a house. Due to the hot weather and malfunctioning
air conditioner, all doors and windows are open. This provides a rousing game of tag, as
both the mouse and the cat are small enough to fit through all doorways and window
frames. Is it possible for the cat and mouse to run through every doorway and wind
frame exactly once? If so, then draw such a route. If not, then prove that such a route is
not possible.
Search online for “house plan diagram” and select a sufficiently detailed plan of a house
for this exercise. Is it possible for the cat and mouse to run through every doorway and
window frame exactly once? If so, then draw such a route. If not, then prove that such a
route is not possible.
Make sure to write up your proof in Claim-Proof form, stating the answer at the beginning
with a claim and using complete sentences and paragraphs in your proof. Be sure to
include any figures that may assist the reader when reading your answer. You may
assume that the reader is familiar with basic graph theory terminology.
Your paper should consist of the following sections:
• Abstract: briefly state the intent of your paper,
• Introduction: state the problem and explain its background,
• Main Result(s): state and prove your result,
• Conclusion: state similar or generalized conjectures that arise from your result.
Assignment 2: Poker Hands
During a 5-card Poker game between five of the most famous (fictional) Poker players,
tension rises when James Bond, Kenny Rogers, and Rusty Ryan each go “all in,” putting
a combined $5 million into the pot. The players reveal their hands to find that
• James Bond has a _____________________________,
• Kenny Rogers has a ____________________________, and
• Rusty Ryan has a ______________________________.
No one wants to let go of any money; each player demands to know the exact likelihood
of each hand; only then can the winner be declared. Since all hands are different, this
will require five separate calculations.
As the dealer, you must determine the winner. Find the general probabilities of each of
the five Poker hands—that is, you must state how likely it would be to get each of the
hands after drawing 5 cards from a 52-card deck (consisting of 13 values, each with 4
suits). Naturally, the hand with the lowest probability wins. It is important that you
prove your answers accurately and concisely, in no more than 2 or 3 pages.
Make sure to write up your proofs in Claim-Proof form, stating the answer at the
beginning with a claim and using complete sentences and paragraphs in your proof.
Write a separate claim and proof for each player’s hand. While you may not need any
figures to assist you, you must use proper notation when referring to combinations and
permutations.
not possible.
Search online for “house plan diagram” and select a sufficiently detailed plan of a house
for this exercise. Is it possible for the cat and mouse to run through every doorway and
window frame exactly once? If so, then draw such a route. If not, then prove that such a
route is not possible.
Make sure to write up your proof in Claim-Proof form, stating the answer at the beginning
with a claim and using complete sentences and paragraphs in your proof. Be sure to
include any figures that may assist the reader when reading your answer. You may
assume that the reader is familiar with basic graph theory terminology.
Your paper should consist of the following sections:
• Abstract: briefly state the intent of your paper,
• Introduction: state the problem and explain its background,
• Main Result(s): state and prove your result,
• Conclusion: state similar or generalized conjectures that arise from your result.
Assignment 2: Poker Hands
During a 5-card Poker game between five of the most famous (fictional) Poker players,
tension rises when James Bond, Kenny Rogers, and Rusty Ryan each go “all in,” putting
a combined $5 million into the pot. The players reveal their hands to find that
• James Bond has a _____________________________,
• Kenny Rogers has a ____________________________, and
• Rusty Ryan has a ______________________________.
No one wants to let go of any money; each player demands to know the exact likelihood
of each hand; only then can the winner be declared. Since all hands are different, this
will require five separate calculations.
As the dealer, you must determine the winner. Find the general probabilities of each of
the five Poker hands—that is, you must state how likely it would be to get each of the
hands after drawing 5 cards from a 52-card deck (consisting of 13 values, each with 4
suits). Naturally, the hand with the lowest probability wins. It is important that you
prove your answers accurately and concisely, in no more than 2 or 3 pages.
Make sure to write up your proofs in Claim-Proof form, stating the answer at the
beginning with a claim and using complete sentences and paragraphs in your proof.
Write a separate claim and proof for each player’s hand. While you may not need any
figures to assist you, you must use proper notation when referring to combinations and
permutations.
Your paper should consist of the following sections:
• Abstract: briefly state the problem and the intent of your paper,
• Introduction: state the basic history and rules of Poker; also define combinations
and probability,
• Main Result(s): state and prove your result(s),
• Conclusion: summarize your work, and make conjectures that arise from your
result(s).
You may choose any 5 of the following non-crossed out Poker hands:
• Royal Flush: The values 10, J, Q, K, A of the same suit.
• Straight Flush: Any 5 consecutive values with the same suit.
• Four of a Kind: All 4 copies of the same value and one additional card.
• Full House: Any 3 copies of one value and any 2 copies of a different value.
• Flush: Any 5 cards of the same suit that do not form a Royal Flush or Straight
Flush.
• Three of a Kind: Any 3 copies of one value and any 2 different values.
• Two Pair: Any 2 copies of one value and any 2 copies of another value and one
additional value.
• Pair: Any 2 copies of one value and any 3 different values.
• High Card: All other Poker hands not previously described.
• Abstract: briefly state the problem and the intent of your paper,
• Introduction: state the basic history and rules of Poker; also define combinations
and probability,
• Main Result(s): state and prove your result(s),
• Conclusion: summarize your work, and make conjectures that arise from your
result(s).
You may choose any 5 of the following non-crossed out Poker hands:
• Royal Flush: The values 10, J, Q, K, A of the same suit.
• Straight Flush: Any 5 consecutive values with the same suit.
• Four of a Kind: All 4 copies of the same value and one additional card.
• Full House: Any 3 copies of one value and any 2 copies of a different value.
• Flush: Any 5 cards of the same suit that do not form a Royal Flush or Straight
Flush.
• Three of a Kind: Any 3 copies of one value and any 2 different values.
• Two Pair: Any 2 copies of one value and any 2 copies of another value and one
additional value.
• Pair: Any 2 copies of one value and any 3 different values.
• High Card: All other Poker hands not previously described.
Halperin et al “Beginning mathematical writing assignments”
Were there any graphs where you expected to find a trail with every edge but didn’t?
_______________ What does this suggest?
Conjecture 3: If within a graph G there exists a trail containing every edge in G, then
G has precisely zero or two vertices of odd degree.
Combining Proposition 2 and Conjecture 3, can we make an even stronger conjecture?
Conjecture 4: A graph G has precisely zero or two vertices of odd degree if and only
if there exists a trail containing every edge of G.
We’ll revisit this topic later in the semester. . .
Homework (due Monday): Writing Assignment
Notes
1Mathematicians use “Claim-Proof” form to state and logically justify an assertion of fact.
2The authors’ tongue-and-cheek approach was inspired by LaRose (LaRose, 1999).
3This idea is courtesy of Crannell (Crannell et al., 2004).
4See Tiny Epic Western or Pandánte.
References
Bahls, P. (2012). Student writing in the quantitative disciplines: A guide for college
faculty. San Francisco, CA: John Wiley & Sons.
Bean, J. C. (2011). Engaging ideas: The professor’s guide to integrating writing, critical
thinking, and active learning in the classroom. San Francisco, CA: John Wiley & Sons.
Crannell, A., LaRose, G., Ratliff, T., & Rykken, E. (2004). Writing projects for mathematics
courses: Crushed clowns, cars, and coffee to go. The Mathematical Association of
America.
Georgia Southern University. (2015). Georgia southern eagles, write! Write! Write! -
georgia southern university quality enhancement plan.
LaRose, G. (1999). Gavin’s calculus projects. Gavin’s Calculus Projects.
http://www.math.lsa.umich.edu/~glarose/courseinfo/calc/calcprojects.html.
Latulippe, J., & Latulippe, C. (2014). Reduce, reuse, recycle: Resources and strategies
for the use of writing projects in mathematics. PRIMUS, 24 (7), 608–625. http://doi.org/
10.1080/10511970.2013.876794
37
Were there any graphs where you expected to find a trail with every edge but didn’t?
_______________ What does this suggest?
Conjecture 3: If within a graph G there exists a trail containing every edge in G, then
G has precisely zero or two vertices of odd degree.
Combining Proposition 2 and Conjecture 3, can we make an even stronger conjecture?
Conjecture 4: A graph G has precisely zero or two vertices of odd degree if and only
if there exists a trail containing every edge of G.
We’ll revisit this topic later in the semester. . .
Homework (due Monday): Writing Assignment
Notes
1Mathematicians use “Claim-Proof” form to state and logically justify an assertion of fact.
2The authors’ tongue-and-cheek approach was inspired by LaRose (LaRose, 1999).
3This idea is courtesy of Crannell (Crannell et al., 2004).
4See Tiny Epic Western or Pandánte.
References
Bahls, P. (2012). Student writing in the quantitative disciplines: A guide for college
faculty. San Francisco, CA: John Wiley & Sons.
Bean, J. C. (2011). Engaging ideas: The professor’s guide to integrating writing, critical
thinking, and active learning in the classroom. San Francisco, CA: John Wiley & Sons.
Crannell, A., LaRose, G., Ratliff, T., & Rykken, E. (2004). Writing projects for mathematics
courses: Crushed clowns, cars, and coffee to go. The Mathematical Association of
America.
Georgia Southern University. (2015). Georgia southern eagles, write! Write! Write! -
georgia southern university quality enhancement plan.
LaRose, G. (1999). Gavin’s calculus projects. Gavin’s Calculus Projects.
http://www.math.lsa.umich.edu/~glarose/courseinfo/calc/calcprojects.html.
Latulippe, J., & Latulippe, C. (2014). Reduce, reuse, recycle: Resources and strategies
for the use of writing projects in mathematics. PRIMUS, 24 (7), 608–625. http://doi.org/
10.1080/10511970.2013.876794
37
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