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Taking Math Outside of theClassroom:Math in the CityPetronela RaduPublished online: 10 Jun 2013.

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PRIMUS, 23(6): 538–549, 2013Copyright © Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970.2013.778381

Taking Math Outside of the Classroom:Math in the City

Petronela Radu

Abstract: Math in the City is an interdisciplinary mathematics course offered atUniversity of Nebraska-Lincoln in which students engage in a real-world experienceto understand current major societal issues of local and national interest. The course isrun in collaboration with local businesses, research centers, and government organiza-tions, that provide data and act as consultants throughout the course. We provide a briefdescription of the course with examples of projects offered over the years. We empha-size service-learning aspects of the program, as well as benefits to students and businesscollaborators.

Keywords: Interdisciplinary course, hands-on learning, project-based learning.

1. Introduction

I liked the fact that we didn’t know what we were capable of doing until we didit. (Math in the City student)

The cold morning of December 2 2011 did not get much notice from the stu-dents of Math in the City, who were getting ready to talk about their workand results on the semester-long projects. The early days of setting up thedata, then creating mathematical models and analyzing them, the long hoursput into preparing the final reports and presentations were all behind them.In their interview attires they were showing their enthusiasm and excitement ofparticipating in this rewarding experience.

These student presentations were given at the yearly Math in the Cityworkshop, where instructors from varied institutions across the United Statescame to learn about an interdisciplinary project-based course. In this program

Address correspondence to Petronela Radu, Department of Mathematics, Universityof Nebraska-Lincoln, Lincoln, NE 68588, USA. E-mail: pradu@math.unl.edu

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students work with local businesses to set-up and analyze models based ondata and activities related to issues of local and national interest. Some of theseprofessors have heard of or participated in similar endeavors, while others wereconsidering it as a new model to teach mathematics. This course came as asolution for many of them in response to the challenge for STEM teachers toprepare the new student generation with a range of skills that will help thementer and rapidly adjust in the workplace after graduation.

The three groups of students took turns talking about their projects that inthe Fall 2011 were done in collaboration with Lincoln’s Finance Department.In the audience there were undergraduates, graduate students, faculty mem-bers, and also, math teachers that were preparing to take the course in Spring2012 as part of their training in the Noyce Program at University of Nebraska-Lincoln (UNL). In this mostly academic group, the “outsiders” were some ofthe business collaborators that have worked with the course over the years,by providing data and guidance for the student projects. The students’ greatperformances illustrated that they learned more than math in this class; theirnewly gained abilities to showcase their work have greatly improved theiremployability skills.

The audience was impressed by the students’ presentations and askedquestions about why certain decisions were taken along the research. Whywas the data chosen this way? How did they decide on the approach to use?The audience also wanted to know more about the student experience (whatwas the most difficult part? what was the most valuable thing you learned?how long did it take to do the research and how long did it take to write thereport?). The business partners commented on the value of their results to thecommunity. Math faculty and students asked and made comments about themathematical background and hypotheses. The overall agreement in the roomwas that such an experience is extremely beneficial to students, so departments,and communities likewise need to learn how to support similar endeavors.

These presentations marked the end of the fifth offering of the courseat UNL, with 13 projects so far and counting. The course is now one of themath capstone courses for students at UNL, but since the background require-ments are relatively minimal (two of the three courses: Differential Equations,Matrix Theory, or Intro to Statistics) students with varied mathematical training(such as non-math majors in their second or third year) had successful expe-riences. The Mathematics Department strives to offer the course every year,so all students have the opportunity to take it at least once during their timeat UNL.

2. WHY MATH IN THE CITY? WHAT IS MATH IN THE CITY?

A strong body of evidence suggests that project-based instruction that includeshands-on learning is an outstanding way to prepare our students for the 21st

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Century (see [11]). Mathematicians, pure and applied, know that teachingmathematics in an interdisciplinary context provides not only the motivation forstudying the material that students need to know, but it immediately connectsstudents with the outside world [5]:

We believe that studying applied math, like learning living languages, providesboth useable knowledge and abstract skills. In math, what we need is quantitativeliteracy, the ability to make quantitative connections whenever life requires (aswhen we are confronted with conflicting medical test results but need to decidewhether to undergo a further procedure) and mathematical modeling, the abilityto move between everyday problems and mathematical formulations (as whenwe decide whether it is better to buy or lease a new car). It is through real-lifeapplications that mathematics emerged in the past, has flourished for centuriesand connects to our culture now.

While this quote comes from an article published in 2011, I started havingsimilar thoughts after teaching my first courses as a newly minted AssistantProfessor at UNL in 2005. In courses such as Differential Equations, I wouldobserve how students struggle with putting word problems into equations andwith creating a mathematical model when a set of data was given (I learnedduring my graduate studies the importance of understanding a mathematicalmodel, see [12]). Thus, in 2006 I created a pilot version of the course Mathin the City, which received strong support from the department and university,as well as from business representatives around the Lincoln area. I taught thecourse several times before my colleague, Stephen Hartke, and I partnered toaddress further development and sustainability issues; these efforts receivedNSF support in the form of a TUES (former CCLI) grant. This support enabledus to create additional course materials, as well as organize a yearly workshopto disseminate the course to other institutions.

Math in the City provides a real-world experience in which studentsengage in hands-on learning and use mathematical modeling to understandcurrent major societal issues of local and national interest. To obtain data andguidance during investigations we seek collaborators among local businesses,research centers, and government organizations, thus creating strong connec-tions between our department (and university) and industry. Throughout thesemester students develop mathematical models based on real data and realis-tic assumptions, so they experience first hand the applicability of mathematicsin society.

Since the first course offering in Spring 2006, we have worked with a widearray of businesses and research centers. Some of the course collaborators havebeen:

● University of Nebraska Medical Center in Omaha – on a project identifyingrisk factors in heart disease;

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● Nebraska Department of Natural Recources – to analyze water levels in thelargest lake in Nebraska, Lake McCounaghy, during a 7-year drought;

● The Schemmer Associates (Lincoln) – who provided data and guidance fora project related to traffic in the city;

● Lancaster County Assessor (Lincoln) – for a statistical analysis of thehousing market during the national housing bubble;

● The Architectural Partnership (Lincoln) – to evaluate costs versus benefits insustainable design;

● The City of Lincoln Recycling, Recycling Enterprises, and Von Busch andSons – to find optimal routes for the recycling trucks;

● The City of Lincoln Finance Department – to perform an analysis of thecity’s financial activities.

The following project description (from Fall 2008) exemplifies the typeand scope of work done in the course; more information on each of the projectscan be found at [9].Sustainable design. Three groups of students analyzed costs versus savingsin sustainable design at two LEED certified buildings: the Nature Center atPioneers Park in Lincoln, NE, and the Prairie Hill Learning Center in Roca,NE. These buildings feature green energy and water-saving features suchas: strawbale walls, geothermal HVAC system, a wind turbine, solar panels,energy-efficient windows and appliances, low VOC emitting paints, cool metalroofs, low flush toilets, sink aerators, etc. One group of students focused on theimpact that these buildings have on the environment by optimizing costs whileminimizing emission of carbon dioxide and other pollutants. All three projectsused linear programming as the main mathematical tool. In each project, stu-dents set-up cost savings functions which included as variables the items or theunits corresponding to a green feature, and the coefficients were taken to be thesavings generated by the particular green feature per unit over a fixed period oftime. The students looked at the best green features in which to invest so thatthe total premium cost paid by the builder would be less than a given amount.For longer periods of time (30 years or more) students also incorporated finan-cial factors such as inflation, increases in energy and water prices, and rate ofreturn for the money, and then analyzed the model under different scenarios forthese factors.

Over the years the students presented their results obtained at two wellattended, yearly events — the University of Nebraska-Lincoln UndergraduateResearch Fair and the Nebraska Research Expo, a conference organized incollaboration with local businesses from Nebraska. The students’ work hasbeen applauded by professors in different disciplines as well as businessrepresentatives.

The course was very well received by the local businesses with which weworked; in fact, there have been more businesses interested in working with this

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program than has been possible to accommodate. As a classroom instructor inacademia it has also been very rewarding to know that two students found jobsafter completing the course as a result of their work in this class. One of thestudents received an internship followed by a job offer from the collaboratinginstitution. A second student used the Math in the City experience to apply fora job that did similar type of analysis as the one done during the course; thecompany hired her after hearing about the work that she had done in Math inthe City.

Math in the City is designed as an interdisciplinary program that allowsstudents to see how mathematics is used outside academia, thus illustratingthe importance of studying mathematics in college. Indeed, as it is pointedout in [2] (which cites [4]) failure to establish relevance is one of the worst10 mistakes in teaching, since:

. . . students learn best when they clearly perceive the relevance of course contentto their interests and career goals. The “trust me” approach to education . . .doesn’t inspire students with a burning desire to learn, and those who do learntend to be motivated only by grades.

A growing body of academic research supports the use of project-basedlearning in schools with technology employed in a meaningful way as a mech-anism to engage students, cut absenteeism, boost cooperative learning skills,and improve test scores; using open-ended projects versus the traditional, directinstruction is far more beneficial [1, 3].

As it has become obvious to teachers of all levels, it is important toconnect problems to practice and the keys to success of this process is tochoose a good problem, implement it well, and verify the results. In Mathin the City this three-step process is done in collaboration with our busi-ness consultants. It is important to recognize that, in the learning processthat takes place, students need to make conjectures, justify them, and checktheir validity. This is more important than “knowing” the answer becauseof the thinking process in which they are involved. Math in the City is ahands-on learning experience that makes the transition from the traditionallecturing style and from the “procedure ...procedure ...procedure ...” teachingapproach to effective and efficient teaching in which the students are activelyinvolved.

In Math in the City the focus is on the project and the application ofthe material; the mathematics is learned and performed in order to solve theproblems. This is in contrast with a standard textbook approach in education,where the applications are often secondary and a sequel to the main material.We believe that solid mathematical education should include both; dependingon students’ future careers this blend may be in different proportions, but webelieve that all will benefit from seeing applications first hand. Also, the course

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provides a bridge between the textbook information learned in other courseswith real-life problems of national interest.

3. CIVIC ENGAGEMENT THROUGH MATH IN THE CITY

Service-learning is a teaching strategy through which students identify, researchand address real community challenges, using knowledge and skills learned inthe classroom. [7]

The course offers an excellent opportunity for students (and more largely,for academia) to get connected with the community around them. Newspaperarticles and TV news become real when students access the data and the infor-mation behind the news. This awareness is usually followed by the students’motivation to understand and tackle the issues that they see as most important.There are several aspects that we believe speak of the civic engagement of thecourse:

3.1. Connection with the Community and the Environment

Students learn during the very first week of classes about the topics and goalsof their projects; in fact, we invite the collaborators to give a presentation (ortwo) about their work and its significance during that first week. As part of thereal-world aspect of the course, students also visit the collaborator’s workplace.When working on routing problems for the recycling companies in Fall 2010,we scheduled a visit to a recycling site, and also visited the Lincoln’s land-fill. It was the volume of recycled (or discarded) material that made a lastingimpression on the students, but maybe even more so, the amount of work andplanning that went into these processes. Most of us did not know that a land-fill site needs a team of engineers and a multi-layered process that goes intodesigning and maintaining a landfill site. At the end of these trips everyonewas committed to the three Rs: Reduce, Reuse, Recycle.

3.2. Connection in Space

Student campuses are mini-universes where the new generations are formed.In these spaces, educators in all disciplines share the belief that we are all con-nected; we know that our students of today will be our doctors and engineersof tomorrow. We have learned that collaborating with our peers will contributeto the progress and development of the outside world. The students in a class,however, don’t always experience this sense that we are all weaving the samepiece. They wonder, “what does my homework have to do with hers?”. In Math

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in the City students choose the project, and in this way they choose each other.I found that in this way students are involved with each other and their work at adeeper level; having a common interest is more motivating to them than choos-ing their teammates based on looks during that first class they have together(something that people at Match.com also figured out!). Once students havea group, they start gaining the confidence that they can do something signifi-cant together; by the end of the project [they gain] “experience working witha group on a much larger project than what could be done by an individual”(again, anonymous quote from a Math in the City student).

I believe that this connectedness is the first level of civic engagement. Theprojects selected from around the city were some that captured the attentionof the local community, or they were the topic in a national front page arti-cle. We looked at water issues in Spring 2006 at a time when Nebraska wasgoing through a 7-year drought; we analyzed the housing market in Lincoln inFall 2006 when the nation was identifying a housing bubble; when the “goinggreen” and sustainable design movements started to gain traction in Nebraskawe looked at sustainable design (projects of Fall 2008); recycling was thetheme when the city started considering a city service for its residents. Forthese large projects students had to identify the issues that mattered, the datathat was needed, the analysis that would be useful to the collaborator. DuringFall 2011 students worked with the City of Lincoln County Assessor to learnabout the city’s financial situation. As we designed the course we expectedstudents to look at interest rates and prioritizing expenses, but one group ofstudents chose to look at the efficiency of police stations. When asked whythey decided this, they gave several reasons that exemplified their clear under-standing of the data, showed the preliminary research they did (quite a largeundertaking in itself), and the feeling of ownership for their project. First, theynoticed that this was the largest expense of the city (it took more than half ofthe budget); second, they took a look at the police stations and they were alittle surprised at their locations throughout the city. They investigated the effi-ciency of these locations and the possibility of adding a fourth location basednot only on financial reasons, but also considering safety aspects for citizens.It was a proud moment for us instructors to see how far students can go with-out any instructor guidance: they contacted over a dozen police stations aroundthe country to assess the efficiency versus staffing of police stations in citiesof similar sizes; they made contact with a local police representative and set-up meetings to discuss these issues (they followed up with her throughout thesemester). When I first designed this course, I never dreamed that studentswould have this level of initiative to carry out a project.

Throughout these projects, students learned to use their communicationskills to solve community problems. They learned that in order to approacha large-scale problem they need to be involved and communicate not onlywith their peers, but also with instructors, business collaborators, and as wementioned they often find resources outside their immediate community. This

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continuous exchange of information between the students, the business, andthe community ultimately benefits all parties.

3.3. Connection in Time

There is also a connection of our students through time. First, we identifythis connection as students of a past project will inform future students’ workthrough their results, analysis, so further investigations (or different ones) couldbe carried out. There is a difference between a former student saying: planahead and an instructor advising the same thing. Students’ perceptions of aninstructor may be altered by the anxiety that comes with someone that assignsgrades, so we are incorporating more ways in which students can learn frompast experiences of students from previous years. We are currently workingon a student handbook that will be handed out to students at the beginning ofa semester. This handbook will contain information on how to best plan anddivide the work, how to deal with challenges, and how to have the best per-formance in the course. The best section, however, may be the one containingadvice from previous students. These are some of the comments that we havereceived so far which clearly show the variety of things that students took fromthe course:

Set goals and map out all of your work for this course as much as possible becauseif you don’t do this the work will creep up on you quickly and it will be easy toget overwhelmed.

Especially in the beginning it might be nice to set up a weekly meeting for yourgroup outside of class even just for 30 minutes so that you all can touch baseand regroup. It is easy to all get going in different directions and it really helpsto have a time where you can all get back on the same page, focus, decide whatremains to be done in the project and figure out how you are going to accomplishit as a group.

Start early and add to the report little by little so when it comes time to the makingthe final report you have a good start.

Leaving yourself enough time to prepare for writing your report presentations iscrucial.

Make sure you have time to meet outside of class and start doing so as soon aspossible. The project is going to take you much more time than you think it will.Also, don’t give in when you want a question answered. If your group doesn’twant to answer it together it is okay for you to work and answer it on your ownand then work to add that to the project as well. You won’t always get along andmany times you will have different ideas, but it’s the work that matters.

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3.4. Deep Learning and Big Dreaming

An important facet of Math in the City is that it promotes deep learning, essen-tial in forming the minds, work skills, and attitudes of tomorrow’s citizens.Thus, an undergraduate degree should be conferred to a student confident inher abilities, so that the importance is not placed on “knowing everything,” butknowing the important ideas and how to go about building on those skills andideas.

[Faculty] must teach for deep learning, realizing that no one can go deep in allareas of content. It is important, therefore, to identify what Wiggins and McTighe(2005) call enduring understanding, those key aspects of a discipline that studentsMUST learn in order to succeed in the next course or, more importantly, in theprofession or as citizens. Deep learning not surface learning is thus essential formastering key concepts and skills in virtually all disciplines. [10]

What students learn in the classroom today is what they will build ontomorrow. More importantly, what they dream today is what they will createtomorrow. If we do not allow students to dream, to practice in the classroom,how much risk will they take when they leave the classroom, enter the work-force and have to be accountable for every decision they make? There is notmuch room for error afterwards and not learning to trust their instincts and fol-low their dreams will be a failure for the education we provided. The safetynet provided by the classroom, teachers, and peers alike will not exist aftergraduation. There will be orders to fill, plans to execute, and often that may hap-pen with little assistance from supervisors or colleagues. Students experiencinggroup work on a real project in college will learn two things: the importanceof building a network of experts and colleagues that you can rely on; and, itis all right to dream and to make (some) errors along the way. For those ofus believing that every challenge is an opportunity for growth, this will be anatural follow-up thought. There is a wonderful freedom that comes with theacceptance that mistakes are part of the process; Yo-Yo Ma said that “your atti-tude has to be, “I welcome that first mistake because now I’m free”” (read thestory behind this quote at [8]).

4. IMPACT OF MATH IN THE CITY

We measure course success through the impact that it has on students, whorecognize the challenges that the course brings, as well as its benefits. Theseare some end-of-semester comments that they had (collected anonymously bythe Bureau of Sociological Research at UNL):

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Have fun and enjoy the experience because although at times the course may seemstressful or like a lot, in the end it really pays off and is a great experience andsomething totally different from anything you’ve ever done in a course before.Being able to get up there and present at the workshop and feel confident in yourresults and your project is the best feeling ever.

How to value different raw data is one of the most important skills I learned.I also gained some skills to pick and choose goals that are reachable from goalsthat cannot be finished.

No specific answer to the problems. Instead of like a normal textbook, you justopen up the back to see if you got it right, in this course you have to decide if youthink it is right or not.

It made you think. A lot.

This is one of the most challenging and most rewarding classes I’ve ever taken.I plan on taking it again next year if possible.

4.1. Benefits to the Students

We summarize here some of the educational and personal benefits that the pro-gram brings to the students. From the beginning, the course was designedso that it enables students to translate a complex real-life situation into amathematical model. Throughout the semester students develop better com-munication skills in writing and for oral presentations through many ways:they write weekly in their journals, they turn in memos to give updates on theirwork, they give slide and poster presentations at the end of the semester, andof course, they turn in a manuscript containing their results and analysis, ina format that resembles an honors thesis. These written manuscripts are firstturned in as drafts for which students receive feedback from the instructors,after which they turn in their final write-ups that will be graded. Guidelines forgrading for each course offering are available on the program’s website at [9].

For every course offering, students worked with some mathematical soft-ware, so they learned SPSS, Maple, Sage, as well as typesetting software suchas LyX and LaTeX. A key aspect remains the exposure to workplaces outsideacademia (they can show off and improve their “employable skills”). On thepersonal level students learn how to deal with setbacks, how to manage theirtime so they can meet deadlines. They learn how to work in groups, and learnto take initiative as instructors have often witnessed it. The entire experienceleads to increased self-confidence and sense of achievement – a great way tostart preparing for the job market. When the average time an employer spendson a resume is about 6 seconds [6], an experience such as Math in the Citywill make the student’s application stand out and increase her chances of beinghired.

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4.2. Other Benefits

The instructors of this class have all noticed how their perspective on teach-ing has changed by the end of the semester; this class showed them howapplying hands-on learning methods increases students’ involvement, so theystarted applying the same techniques in other courses they teach. Personally,after seeing the benefits of writing and oral presentations, I started assigningmore essays, in lower-level classes, such as Differential Equations, as well asin advanced graduate courses. I can think of a few ways to increase a graduatestudent’s employability, but probably none is faster done than asking studentsto give oral presentations and training them to showcase their work. A graduatestudent in the department spent a good portion of her successful job interviewtalking about Math in the City, sparking interest from the hiring institution.

Finally, but not of least importance, we mention the benefits that the coursebrings to the sponsor. Our collaborators have found that the course is a greatavenue to advertise to the public some of their programs. Since the studentsoften end up giving presentations on their work, a potentially large numberof people find out information about these activities (for example recycling,sustainable design, city’s efforts to maintain financial stability). Students oftenfound ways to improve or optimize the activities of the business collabora-tor; they found better ways for the routing of recycling trucks, they studiedand made suggestions to improve the efficiency of the police stations, theyformed models to predict water levels in Lake McCounaghy given weather andprecipitation conditions.

The course is all in all a win-win-win situation for students, instructors,and collaborators.

REFERENCES

1. Boaler, J. 1998. Open and closed mathematics: student experiences andunderstandings. Journal for Research in Mathematics Education. 29(1):41–62.

2. Dewar, J. 2009. How should we respond when students ask: when will weever use this? Association for Women in Mathematics. 6: 20–22.

3. Edutopia. What works in education. http://www.edutopia.org/project-based-learning-research. Accessed May 2013.

4. Felder, R. M. and R. Brent. 2009. Tomorrow’s professor, message #961:the ten worst teaching mistakes. http://mailman.stanford.edu/mailman/listinfo/tomorrows-professor. Accessed May 2013.

5. Garfunkel, S. and D. Mumford. 2011. How to fix our math educa-tion. http://www.nytimes.com/2011/08/25/opinion/how-to-fix-our-math-education.html?r=1. Accessed May 2013.

6. Giang, V. 2012. What recruiters look at during the 6 seconds they spend onyour resume. http://www.businessinsider.com/heres-what-recruiters-look-

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at-during-the-6-seconds-they-spend-on-your-resume-2012-4. (Accessed 9April 2012). Accessed May 2013.

7. Kids consortium. http://www.kidsconsortium.org/learningservice.php.Accessed May 2013.

8. Lehrer, J. ‘Imagine’ that: fostering creativity in the workplace. http://www.npr.org/2012/03/21/148607182/fostering-creativity-and-imagination-in-the-workplace. Accessed May 2013.

9. Math in the City. http://www.math.unl.edu/~math-mitc/. Accessed May2013.

10. Mills, B. Promoting deep learning. http://theideacenter.org/sites/default/files/IDEA_Paper_47.pdf. Accessed May 2013.

11. Partnership for 21st century skills. 2005. Assessment of 21st cen-tury skills: the current landscape. http://www.p21.org/storage/images/stories/otherdocs/Assessment_Landscape.pdf. Accessed May 2013.

12. Radu, P. 2011. Math in the city – an example of hands-on learning. TheBIG Notebook–BIG SIGMAA newsletter. April: 5–8. Accessed May 2013.

BIOGRAPHICAL SKETCH

Petronela Radu received her Ph.D. from Carnegie Mellon University,Pittsburgh, PA. She joined the Department of Mathematics at Universityof Nebraska Lincoln as an Assistant Professor in 2005. She is conductingresearch in applied mathematics and engaging students in solving problemswith applications in the real world.

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