Beginning Mathematical Writing Assignments
BBeginning Mathematical WritingAssignments
Alexander Halperin, Colton Magnant, Zhuojun MagnantOctober 27, 2018
Abstract
Writing assignments in any mathematics course alwayspresent several challenges, particularly in lower-level classeswhere the students are not expecting to write more than a fewwords at a time. Developed based on strategies from severalsources, the two small writing assignments included in thispaper represent a gentle introduction to the writing ofmathematics and can be utilized in a variety of low-to-middlelevel courses in a mathematics major.
It is often a struggle to get mathematics students to write much ofanything beyond their name. In fact, many math majors specificallychoose mathematics believing they can avoid writing altogether.When faced with the brutal reality that writing in mathematics isjust as important and in fact far more detail-sensitive than in mostother disciplines, many students seriously consider changingmajors. The problem is cyclic; many algebra, precalculus, andcalculus classes require little to no mathematical writing, eitherthrough problem-solving, exposition, or even self-reflection afteran assignment. Students, in turn, become accustomed to this lackof writing and associate mathematics with poor (at best)explanations.The short writing assignments contained in this work togetherrepresent our attempt at a solution to several problems at once.First, these assignments provide a gentle, mathematical baby-stepinto the formal writing of mathematics. Second, they were used aspart of a larger research project in which the authors comparedthe progress of students from two different classes at two differentuniversities, one class with an intervention of a larger writingassignment [6] and the other without. We omit the details of this1arger assignment as our focus here is on the smaller writingassignments. Third and related to the previous item, theseassignments, with minimal background introduction, can beassigned in virtually any course within a math program. In short,they were developed as both a measurement tool and a gentleintroduction to formal mathematical writing.These writing assignments were developed using a combinationof strategies including ideas from Walvoord [11], Bahls [1], Bean[2], and Crannell et al. [4] as well as ideas stemming from our ownpersonal experience in taking and teaching courses in mathematicsover the years. The assignments were intended to foster effectivewriting habits and, at the same time, develop students’ skills in theareas of argumentation, analysis and synthesis. In order to provetheir claimed solution, the students must argue using credibleevidence and supporting logic. Effective analysis and description ofthe situation in both questions along with their correspondingdifficulties is critical to a successful complete solution. Finally, anelement of synthesis is expected in the summarizing conclusion,where the students must consider a possible “natural” next step asa direction for future work.Both of the following writing assignments were assigned in eachof two different classes, one at each of two different universities.University A is a mid-sized public regional comprehensiveuniversity while University B is a large public regionalcomprehensive university. At University A, the assignments wereused in an introductory Discrete Mathematics course. At UniversityB, the assignments were used in a course on introduction to proofscalled Mathematical Structures. Each class had about 30 students,primarily second-year undergraduates with a handful of first-yearand third-year students as well. Most of these students had neverwritten more than a sentence or two in a math class. Both classeswere learning Claim-Proof form of mathematical writing so anadditional goal of these assignments was to reinforce this writingstyle.For both of the writing assignments, students spent a day inclass solving a related problem so that outside of class, they couldfocus almost exclusively on the writing component rather thandwelling on the mathematics. The mathematical content of theassignments was also deliberately involved — students needed tospend the entire period understanding and answering the problem— but not terribly difficult to further encourage the students tofocus on the writing. Classroom discussions centered around the Assignment 1: Cat and Mouse
A cat chases a mouse in and out of a house whose floor plan isshown below. Due to the hot weather and malfunctioning airconditioner, all doors and windows are open. This provides arousing game of tag, as both the mouse and the cat are smallenough to fit through all doorways and window frames. Is ispossible for the cat and mouse to run through every doorway andwindow frame exactly once? If so, then draw such a route. If not,then prove that such a route is not possible.Figure 1: Image from http://zazio.xyz/maison-modernes/plan-de-maison-entunisie-100m2.htmlMake sure to write up your proof in Claim-Proof form, stating theanswer at the beginning with a claim and using completesentences and paragraphs in your proof. Be sure to include anyfigures that may assist the reader when reading your answer. Youshould motivate your result and define all necessary terminology.Expect to write about a page of typed text. Your final work will bescored using the “Writing rubric” posted on the course website.Your paper should consist of the following sections:• Abstract: briefly state the problem and the intent of yourpaper,• Introduction: define relevant mathematical concepts andbriefly discuss this question and how it relates to the Bridgesof Königsberg problem (as we discussed in class),4 Main Result(s): state and prove your result(s),• Conclusion: summarize your work, and make conjectures thatarise from your result(s).
During a 5-card Poker game between three of the most famous(fictional) Poker players, tension rises when James Bond [3], KennyRogers [10], and Rusty Ryan [9] each go “all in,” putting acombined $5 million into the pot. The situation resemblessomething like this:Figure 2:
His Station and Four Aces , C. M. Coolidge, 1903.The players reveal their hands to find that• James Bond has a ______________________________,• Kenny Rogers has a ___________________________, and• Rusty Ryan has a _______________________________.Of course, no one wants to let go of any money. In fact, each playerdemands to know the exact likelihood of each hand; only then canthe winner be declared. Since each player has a different hand,this will require three separate computations.As the dealer, you must determine the winner. Find the generalprobabilities of each of the five Poker hands—that is, you muststate how likely it would be to get each of the hands after drawing5 cards from a 52-card deck (consisting of 13 values, each with 45uits). Naturally, the hand with the lowest probability wins. It isimportant that you prove your answers accurately and concisely, inno more than 2 or 3 pages.Make sure to write up your proofs in Claim-Proof form, stating theanswer at the beginning with a claim and using completesentences and paragraphs in your proof. Write a separate claimand proof for each player’s hand. While you may not need anyfigures to assist you, you must use proper notation when referringto combinations and permutations.Your paper should consist of the following sections:• Abstract: briefly state the problem and the intent of yourpaper,• Introduction: state the basic history and rules of Poker; alsodefine combinations and probability,• Main Result(s): state and prove your result(s),• Conclusion: summarize your work, and make conjectures thatarise from your result(s).You may choose any three of the following (non crossed-out) Pokerhands:•
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 oneadditional card.•
Full House:
Any 3 copies of one value and any 2 copies of adifferent value.•
Flush:
Any 5 cards of the same suit that do not form a RoyalFlush orStraight Flush.•
Straight:
Any 5 consecutive values that do not form a RoyalFlush or a Straight Flush. 6
Three of a Kind:
Any 3 copies of one value and any 2different values.•
Two Pair:
Any 2 copies of one value and any 2 copies ofanother 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.
The playful nature of both questions is intended to welcome,rather than intimidate, students when they first read theassignment in class. Immediately after reading the problem,students receive a handout with questions pertaining to the paper.For the rest of the class period, students work in groups todiscover a solution. The instructor tours the classroom, answeringminor questions when necessary. We detail this approach for eachassignment in the subsequent sections.Students collaborated during class but were expected to eachwrite their own separate paper, as opposed to Latulippe andLatulippe’s assignments in [14], in which 2–3-student groupsturned in a single essay. This was to ensure that each student wasresponsible for their own writing, as we prioritized writing overproblem-solving. (Although the QEP rubric weights math andwriting equally, most students understood the solutions to bothwriting assigments by the end of the lecture, meaning that theirmath scores should have been high with little variance.) For the “Cat and Mouse” assignment, students first explore thefamous “Seven Bridges of Königsberg problem” [8], in which atraveler tries, in a continuous route, to cross each of seven bridgesin Königsberg, Prussia exactly once. After some trial and error,students discover that such a route is impossible, along with therealization that the lack of solution must be proved , rather than asserted . Listed below the problem are the steps to the proof, outof order , for students to rearrange. At the end of the exercise, theclass recaps the argument to the instructor, in the students’ ownwords. From there, the students have the necessary mathematical The purpose of the poker writing assignment is to again showstudents that mathematics is primarily carried out in words, ratherthan symbols. Further, students should understand the power ofthe “combinatorial proof,” in which quantities are counted usingbasic multiplication, factorials, and combinations. A problem thatcould take hundreds of algebraic calculations by brute force cansometimes be answered in a few short sentences in acombinatorial proof, hence making it the far more desirable option.The writing assignment on poker hands comes at the end of aweek’s worth of combinatorics lessons. Students have studied the combination “ n choose r ,” or nCr , and learned its combinatorialdefinition (“ nCr is the number of ways to choose a subset with r elements out of a set with n elements”) as well as derived itsalgebraic formula ( nCr = n ! / ( n − r )! r !). Further, they have dealt withcounting problems, including a poker hand example and severalmore involving playing cards. Most importantly, students havelearned to solve these problems by viewing nCr through itscombinatorial definition, which emphasizes exposition andconceptual understanding over calculation. Thus, the algebraic8ormula is more a technical result and is only used at the end of aproblem to find an exact numerical answer.After receiving the poker hands assignment at the beginning ofclass, students spend the rest of the period working in groups tocount the number of each type of poker hand displayed on thesecond page (including the crossed out hands). The professorroams the classroom, sorting out any misconceptions andcorrecting what are usually small errors. By the end of class, allstudent groups have counted all or nearly all poker hands.The main task for their writing assignment is for students toformally write their ideas in class as a logical sequence of steps, inwhich they reformulate a poker hand into the values and/or suitsthat are chosen to form it. Of course, they still must providebackground for this problem, but that is usually simpler than theprevious writing assignment. Most of them find the poker handsituation more gripping than the cat-mouse chase, and many areeager to research the history of poker (and, in some cases, discussit at great length). In fact, some students took such interest withthe history of poker that it dominated the Introduction. To remedythis, future assignments will specify a five-line limit to the “history”portion of the Introduction, and the rubric will be adjustedaccordingly.This being the second assignment, the instructors were able todiscuss the issues with audience awareness so the scores in thisarea were slightly improved over the Cat and Mouse assignment.Other areas where the students showed a bit of weakness wasparagraph structure, transition between paragraphs, and overallflow. In particular, several students listed calculations for the threechosen hands with almost no discussion in between. One way tocombat this misunderstanding may be to create a different rubricand provide it to students with the assignment. We discuss this atthe end of the next section and give such a rubric in the Appendix. Our modest goal was to use these writing assignments simply as anintroduction to mathematical writing, opening the door to theworld of written mathematics. That said, given the ease in whichstudents were able to state their solutions, we believe we shouldadd a degree of difficulty to each assignment. For the cat andmouse assignment, this could mean asking for a more generalproof that any multigraph with more than two vertices of odddegree has no Eulerian trail. For the poker assignment, we areconsidering requiring students to find the probability of a “high9ard,” (the most difficult to explain), including a wild card (i.e., acard that can take on any value or suit), or perhaps a variant gamewith a different number of card values and/or suits. In the future, we hope to further incorporate a small reflectionpiece after each assignment in which the students will reflect ontheir process of writing, which may (should) include revision, peer-review, editing, etc. We hope to use this information to enlightenstudents to the effective writing processes that not only make themstronger writers but also, more immediately, result in bettergrades. Although explicit mention of Process writing was notincluded in the assignment prompts, it was discussed in class,particularly when discussing outlines, revision, and peer review.We intend to include more deliberate details about Process Writingin future iterations.Broadly speaking, students show better performance when theaudience for the assignment is clear, supporting the theory in theMAA Instructional Practices Guide [13, p. 91]. Admittedly, thespecific audience was omitted from these assignment promptssince we stated in class that the intended audience was theirpeers, both inside and outside the course. This audience waspurposely chosen to encourage peer review in the revision process.We were pleasantly surprised at the amount and effectiveness ofthe peer reviews and intend to be more intentional about theaudience in further iterations of these assignments, in particular,including explicit statements about audience in the assignmentprompts.We also plan to develop more advanced and challengingproblems to further the students’ writing experience later in theirmathematical careers. In both of these assignments, althoughstudents spent a day on each of the Seven Bridges of Königsbergand poker hands, the actual mathematical solutions were short, nolonger than a paragraph each. Although our students appreciatedthe succinctness of a well-written mathematical proof, we do wantthem to have experience with writing a proof that, by itself, spansat 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 combinatorialquestion. While similar to the approach used in [5], we wouldrequire students to find a question outside of class instead of citingprevious course material. Since, unlike Pinter, our classes containexclusively math majors and minors, we believe the addedcomponent of researching combinatorial problems outside of classwould be a fair, if not challenging, additional requirement. Such an
Tiny Epic Western or Panda´nte . References [1] Bahls, Patrick
Student Writing in the Quantitative Disciplines .John Wiley & Sons, 2012.[2] Bean, John
Engaging Ideas . John Wiley & Sons, 2011.[3] Craig, Daniel, act.
Casino Royale . Columbia Pictures, EonProductions, Casino Royale Productions, 2006. Film.[4] Crannell, Annalisa and LaRose, Gavin and Ratliff, Thomas andRykkens, Elyn
Writing Projects for Mathematics Courses .Mathematical Association of America, 2004.[5] Pinter, Mike Writing to Enhance Understanding in GeneralEducation Mathematics Courses,
PRIMUS , 24:7, 626–636,2014.[6] Flateby, Teresa and Magnant, Colton and Nasseh, Saeed
Mathematical writing assignment for deeper understandingand process writing . NILOA - DQP Assignment Library, 2016.[7]
Georgia Southern Eagles, Write! Write! Write! - GeorgiaSouthern University Quality Enhancement Plan . GeorgiaSouthern University, 2015.[8] Paoletti, Teo “Leonard Euler’s Solution to the K¨onigsbergBridge Problem.” MAA Press, 2011.[9] Pitt, Brad, act.
Ocean’s Eleven . Warner Bros., VillageRoadshow Pictures, NPV Entertainment, 2001. Film.[10] Rogers, Kenny.
The Gambler . United Artists, 1978. CD.1111] Walvoord, Barbara
Assessing and Improving StudentWriting in College . John Wiley & Sons, 2014.[12] LaRose, Gavin
Gavin’s Calculus Projects . , 1999.[13] Mathematical Association of America MAA InstructionalPractices Guide
Mathematical Association of America, 2018.[14] Latulippe, Joe and Latulippe, Christine Reduce, Reuse,Recycle: Resources and Strategies for the Use of WritingProjects in Mathematics,
PRIMUS , 24:7, 608–625, 2014.12 riting Rubric Excerpt
Trait Does notmeet (1) Attempted(2) Approaches(3) Meets (4) Exceeds(5)AssignmentRequirements
The writeris off topicand/oromits mostor all of theassignmentrequirements. The writeraddressestheappropriate topicbut onlysuperficially addressesthe as-signment requirements. The writeraddressesthe ap-propriatetopic andmeets theassignmentrequirements. The writeraddressesthe ap-propriatetopic andclearly andcorrectly fulfills eachaspect of the assignment requirements. The writeraddressesthe ap-propriatetopic andclearly,correctly,andconcisely fulfills eachaspect of the assignmentrequirements.
Reasoning(proof)
Thelogical connec-tion of the argumentis weak,leaving theargumentorexplanationunclear. A“proof byexample” falls here. Thereasoningoffersapparentsupportfor theargument,but theargumentorexplanation is weak. Collectively,the logicoffers ad-equate supportfor theargument,but theargumentorexplanationremainsunclearor incomplete. Collectively,the logicsupportsandadvancestheargumentorexplanation of theproof. Collectively,thelogical stepsoffer compelling support which clearlyadvancestheargument or explanation of the proof.
Quality ofDetails
Details aresuperficialor do notdevelop theproof. Detailsarelooselyrelated tothe proof.Many do notprovide supporting Detailsarerelated tothe proofbut incon-sistently provide supporting statements,credible Details provide supporting statements, credible evidence,or theexamples necessaryto explain Compellingdetails provide supporting statements,credible evidence,or the ex-amples necessaryto explaintatements,credible evidence,or the ex-amples necessaryto explainorpersuadeadequately. evidence,or the ex-amples necessaryto explainorpersuadeadequately. orpersuadeadequately. orpersuadeeffectively.
Math 210: Discrete Mathematics Fall 2017
Lecture 1: The Königsburg Bridge Problem
Instructor: Date: 8/28/17
Problem 1.
Consider the following layout in Königsburg, Prussia:Is there a route through the city that crosses each bridge exactly once?
There are two ways we can try to figure out the answer:1. If we think such a route exists, then .2. If we think such a route does not exist, thenWhich method is easier? Does that mean that method is correct? __________________________sing the maps on the next page, try the easier method for a fewminutes, and see what you get.Try several different routes until you get a solution, or until you think asolution doesn’t exist. ecture 1: The K¨onigsburg Bridge Problem ecture 1: The K¨onigsburg Bridge Problem Answer to Problem 1: . (Euler, 1736)To see this, we can view the map of K¨onigsburg as a graph . Eachland mass is represented by a vertex , and each bridge isrepresented by an edge . A route through the graph that uses anedge at most one time is called a trail .The steps of the claim and proof are posted below, but in thewrong order. Unscramble the steps to obtain a complete andcorrect claim and proof of the problem.1.
Proof:
2. Call this graph G . Hence, we see that G consists of vertices and edges.3. Claim:
4. There is no route through K¨onigsburg that traverses everybridge exactly once.5. As a result, there cannot exist a trail in G that contains everyedge of G .6. It now suffices to prove the claim, “There is no trail in G thatcontains every edge of G .”7. However, each vertex in G touches an number of edges.8. Hence, if T contains every edge in G , then T must have atleast two vertices that touch an even number of edges in T . ecture 1: The K¨onigsburg Bridge Problem T touches an (circle one) even/odd number ofedges in T .11. We represent the map of Königsburg with a graph in thefollowing way: draw a vertex for each land mass and an edge for each bridge.12. This is because each middle vertex in T is entered by oneedge and then exited by another.Using the above reasoning, we can generalize the answer to Problem 1 as follows: Proposition 2.
Using Proposition 2, which of the following graphs might contain atrail with every edge? For those that cannot, can you succinctlyexplain why? For those that might, can you find such a trail?Were there any graphs where you expected to find a trail withevery edge but didn’t? . What does this suggest? ecture 1: The K¨onigsburg Bridge Problem Conjecture 3.
Combining Proposition 2 and Conjecture 3, can we make an evenstronger conjecture?
Conjecture 4.
We’ll revisit this topic later in the semester...
Homework (due Monday):
Writing Assignment oker Paper Grading Rubric
Note that all four criteria in the Main Results section applyto each of your five main results, hence the “×5” in theScore section.
Section Standard ScoreAbstract
Restate the problemState the paper objective /5/4State problem-solving methods used /1
Introduction
Provide a brief history of poker, at most 5 linesDescribe the rules of poker /10/10Restate player hands /10
Main Results
Use Claim-Proof formAccurately find probability /2 × 5/3 × 5Write clearly and correctly /4 × 5Utilize ( /1 × 5
Conclusion
Summarize resultsState a new question /4/4State another new question /2