Since starting my career as a faculty member in 2003, I jumped right in to K-12 Outreach and have never looked back. I was motivated by my strong connection to my community, which is located in St. Lawrence County, a geographically isolated, rural part of upstate New York. All K-12 districts in this county share the same problems of limited resources, significant poverty rates, and a “high needs” population. My choice to become involved in K-12 Outreach was a personal one. ** **I had a very nonlinear path to becoming a mathematician. I was raised by a single mom who sold cars and told me I could do anything I wanted to if I hunkered down and worked hard. I went to three different colleges, changed majors three times, and took five years to get my undergraduate degree—waitressing for the last three years to support myself. I only had one female math teacher in 8^{th} grade and one female math professor—but not until graduate school. My point is, I didn’t have many female STEM role models, but honestly not much of this occurred to me until I started to get involved in K-12 Outreach. However, I quickly understood that these experiences are not the norm and that not every child has an encouraging support system to motivate them. Even for students who do have strong family support, a lack of opportunities for resume building activities or enrichment such as Robotics or Science Olympiad or even an AP Physics class means they are not even competitive when they apply to colleges. I am raising two daughters in this community—they and their peers deserve the same opportunities as students in affluent suburbs scattered across “downstate” New York and elsewhere.

Feedback I’ve received from faculty from a variety of Universities that do K-12 Outreach imply that a common thread is a feeling of wanting to “give back” or to honor a K-12 teacher that made a difference in their lives. The bottom line is that this sort of service to the broader community is a win-win situation. In times of major budget cuts in education, new curriculum and assessments, exhausted teachers, overworked parents, and a new generation of students who need STEM problem solving skills more than ever, it feels great to help out in any possible way. In this article, I’ll describe what K-12 Outreach is and share examples about how mathematics faculty can get involved on a variety of levels. My hope is that, as mathematicians, we can share our expertise with and also learn from the K-12 community to strengthen STEM education through collaboration.

I consider K-12 Outreach to be a partnership with local school districts to improve education, to provide unique learning experiences for everyone involved, and to work collaboratively towards building a future generation of problem solvers. Although this is a broad definition, it allows for a wide range of activities that can help achieve those objectives. The key component for K-12 Outreach is the *partnership*, making genuine connections with superintendents, teachers, principals and students. Approaching the partnership with an understanding that each person has a critical expertise can make a K-12 program a success. I have had some efforts succeed and some that were epic fails, and both relied on trust and appreciation of all the people involved.

The first thing I always do when forming a new partnership is admit that I am by no means an expert (or even qualified) at teaching school-aged children. I rely on teachers to help me understand the appropriate level of material, identify challenging topics that could use more relevance and motivation, and to communicate in a language students can grasp. I remember leading a session about drawing a scaled roller coaster blue print to a group of 7^{th} and 8^{th} graders at our summer camp one year and thinking it was going well. Then I was met with blank stares and nobody knew how to get started. Luckily there was an 8^{th} grade teacher in the room who reworded all my directions for them and they immediately got to work.

Likewise, teachers often do not have the resources or time to learn about how mathematics is being used to solve real world problems. Math modeling, open-ended questions, and interdisciplinary problems within the mathematics classroom are rare, yet emerging, scenarios in K-12 schools. From my experiences, teachers need to be able to trust and feel comfortable asking questions with faculty. When running teacher professional development workshops, I usually have undergraduate and graduate student helpers. They make the setting more comfortable and bridge the gap in terms of technical expertise. In general, I found that using college students can strengthen any level of K-12 Outreach. They usually have great energy and insight and are closer in age to the students we are serving. Participating also is a way to strengthen their resumes and instill an understanding in future mathematicians that K-12 Outreach is valuable.

I have piloted some small-scale efforts as well as participated in both state and federally funded student driven and teacher professional development programs. One effort simply involved organizing an essay contest for local middle school students to celebrate Math Awareness Month (April). Our math club spearheaded the whole thing. We got local businesses to donate prizes (camping equipment, gift certificate to a music shop, sporting equipment) and then students had to relate their essay to how mathematics is used in that arena (for example, why is mathematics important if you are planning a camping trip or how is mathematics used in baseball?) All we really had to do was circulate an announcement to superintendents and then enjoy reading the essays and choosing winners.

There are a variety of pre-existing national STEM programs that provide ways for faculty to make connections with K-12 teachers and students. MATHCOUNTS is a middle school competition that I have been the local director of for the last 11 years. We provide the facilities to hold the annual competition and Clarkson Student volunteers have even worked with teachers throughout the school year to coach teams. In the past, we have also provided one-day workshops for teachers to help them develop coaching activities.

Another opportunity that requires no funding (and actually provides an honorarium!) is to become a judge (or problem author) for the SIAM (Society for Industrial and Applied Mathematics) Moody’s Mega Math (M3) Challenge. The M3 Challenge is a free mathematical modeling competition for high school juniors and seniors held annually in March. Judges have a week to read through roughly forty solution papers online and score them based on a given rubric. See __http://m3challenge.siam.org/__ for more information.

A much more ambitious program is our NYSED STEP (Science Technology Entry Program) after-school and summer camp program, called IMPETUS for Career Success (Integrated Math and Physics for Entry to Undergraduate STEM). This program connects Clarkson faculty and graduate/undergraduate students with 11 school districts and 150+ 7-12 grade students. Highlights are a week-long Summer Roller Coaster Engineering Camp, weekly after-school STEM enrichment activities which include research projects, a model roller coaster design competition, tutoring and mentoring services, and monthly on-campus STEM workshops centered around research and STEM careers, providing a variety of STEM experiences. At camp, students apply math, physics, and simulation with hands-on lab activities. Students predict the behavior of a roller coaster traveling along a wall-mounted track whose shape can be adjusted to accommodate multiple hills, loops, and jumps. Activities include designing a roller coaster from a scaled drawing and wire model that undergoes a complete energy and safety analysis and is simulated via software so students experience their ride virtually and cross-check their velocity and acceleration computations. We also use the VR2002 Virtual Roller Coaster to teach students about accelerations, model predictions, and data analysis. Students visit a Six Flags to collect acceleration data wearing kinematic vest. The highlight is a workshop with a roller coaster engineer who built The Comet. To see more about the scope of this program, see __http://web2.clarkson.edu/projects/impetus/__.

Getting started in K-12 outreach activities seems intimidating but the pay-off is huge. A simple web search will reveal numerous programs, examples, curriculum samples, and funding opportunities that may seem overwhelming. The most important step is getting started and then to keep trying and learning. For my roller coaster camp program, we were denied funding twice before finally being awarded a NYSED grant. Seeing the evolution from the birth of the idea to where we are now is one of the most rewarding experiences of my career. Better yet is seeing what our graduating seniors go on to do. We are in our tenth year and the program is continuously changing and improving. My advice is to start small but think big.

]]>*Editor’s note: The editorial board believes that in our discussion of teaching and learning, it is important to include the authentic voices of current and former undergraduate students reflecting on their experiences with mathematics. *

When I graduated from Vassar College in 2010 with degrees in math and Italian, I wasn’t sure what was next for me. I applied for math-related jobs at my favorite media companies. Ultimately, Time Inc. offered me a position as a Data Analyst, a job which has been an ideal blend of my mathematical and entertainment interests. I manage store-level distributions for three magazines, Us Weekly, Rolling Stone, and Men’s Journal, all published by Wenner, a primary client. I determine how many copies of every issue go into each store by using formulas based on the store’s available checkout pockets and average sales. At Time Inc., I have been impressed and surprised by the variety of math-related projects. There is a Shopper Insights group that has developed an eye-tracking system that follows the movement of a consumer’s pupils while shopping and helps optimize magazine placement in stores. The Research divisions work on projects that include using subscriber data to help expand the reach of our brands and analyzing historical data to create new pricing strategies. They are doing a zone- pricing test for People magazine, where they are removing the cover price and setting different prices for different regions. In this blog post, I use examples from my work experience over the last five years to suggest ways in which undergraduate mathematics majors can be better prepared for math-related positions in companies. I discuss how I wish I had learned more about applications, computer science, statistics, and connections to other STEM fields.

*Applications*

I wish that I had been introduced earlier and more often to applications, as they would have provided me with a better idea of potential areas of specialization after graduation. For example, in linear algebra we could have learned about the role eigenvectors play in Google’s PageRank algorithm, and in number theory we could have learned about how encryption facilitates e-commerce. My textbooks and courses were mostly filled with theorems, definitions, and proofs, and relatively few examples of applications. With more such examples, I believe that students would think more about the value of a math degree and the growing demand for graduates with a math major. Vassar has recently begun inviting graduates back to talk about their career paths. I wish this program had existed when I was there. I would have also liked to learn more about fields where we are only just starting to discover the prominence of math, such as web development and social networks. Additionally, there are many industries in which math’s long-existing role continues to expand, such as movie animation and national security. Incorporating examples like these into the curriculum shows students that mathematical theories influence new applications, and in turn, new applications drive theoretical research by uncovering additional problems. Perhaps an introductory course focusing on real-world applications (with each unit dedicated to a different field where math is used) could show students more of these connections.

*Other STEM fields*

The mathematical sciences continue to be the foundation for exciting research and development in the other STEM fields. Yet I’m sure there are other math graduates like me who didn’t take classes on these subjects and were surprised at the extent to which these fields are used in their lines of work. Much of the work in rapidly evolving areas such as compressed sensing or drug design is being done by people with a foundation in multiple STEM disciplines. To keep up with the broadening of the mathematical sciences and to equip students for a wider range of careers, I think a requirement to take a course in at least one other math-related discipline would be an asset to majors. Students would also benefit from improved interdepartmental collaboration, which could include joint courses that count for credit in more than one discipline or classes co-taught by professors from different departments. For example, a class using computer science skills to analyze large data sets could be applied towards either a computer science or math major. Freshmen and sophomores would take comfort in knowing that, regardless of which subject(s) they pursue further, their math departments offer many worthwhile options beyond core math classes. I might have double majored, or at least taken more science and technology-based courses, had an environment more like this existed.

*Statistics*

I enjoyed the variety of math courses that I took (e.g., linear algebra, modern algebra, multivariable calculus, number theory, probability, and real analysis), yet I wish I had selected my courses with more regard to post-college interests. If I could redo my undergraduate years, I would take more statistics courses. I think a department requirement would help students recognize how important a statistics background is in increasing their mathematical value, and by extension, their employability in data-driven careers. Without taking statistics, students who end up in mathematical jobs would likely have to teach themselves key concepts and tools, such as modeling via simulation or statistical inference, in the workplace. More data than ever is generated, collected, and used for research in today’s world. Because of this, fields from neuroscience to advertising are looking for employees with statistical and computational expertise. The Mathematical Sciences in 2025 says that by 2018, U.S. businesses will need another 140,000-190,000 employees with advanced quantitative skills and deep analytical talent, and who are adept in working with big data. Courses where students work with large data sets and form their own conclusions would be beneficial. I don’t use high-level statistics in my job, but I use many tools that are honed in a statistics class. My work revolves around organizing, analyzing, interpreting, and presenting data. If I had studied some more advanced statistics concepts, I might have been able to find ways to apply them to my job. For example, reading about hierarchical models in The Mathematical Sciences in 2025 made me wonder if my company uses them, especially since it seems like they might be valuable in determining magazines’ sales potential. With the print magazine industry struggling, we analyze data to try to find ways to cut costs without sacrificing revenue. I sift through the numbers to find anomalies, opportunities, and trends, and I use the data to generate ideas for tests and measure results. I also have to make the best of messy data. For instance, our second largest wholesaler recently went out of business, so the twenty thousand stores serviced by that wholesaler were suddenly without magazines until the chains made deals with new wholesalers. Unsurprisingly, the sales data from that transitional period is one big anomaly.

*Computer Science*

I also wasn’t aware of the extent to which math is used in computer science, and of how vital computer science is to ongoing developments in countless fields. I didn’t even consider enrolling in a computer science course. Looking back, I wish that I had been required to take courses in that department. Having knowledge in areas that combine math and computer science skills (e.g., math modeling, simulations, programming, and coding) has become more essential for mathematical careers. Although I don’t use advanced computer science concepts in my job, I’ve had to learn certain data analysis and computer systems skills that I wish I had gotten a head start on in college. I was initially surprised to see that Time Inc. has its own systems built by in-house programmers that hold data on each magazine-selling store in the country and suggest a number of copies to put in each store. I have taught myself rudimentary coding to better understand the logic, structure, and language, but I would have loved to have had a jumpstart in college on querying data and forming conclusions. I communicate frequently with programmers, testing systems and making suggestions for enhancements. Recently, they were having trouble getting a report to display necessary results, and I gave them a query I had written which helped them finish building the report. I was able to figure out the logic in this instance, but building queries for other reports would be beyond my understanding. There have been numerous situations like this where I’ve felt that having even a basic computer science foundation could have led to faster progress and a stronger group effort.

Much of 21st century research will have a foundation in math, and there will be surprising connections to other fields, as well as jobs that haven’t even been conceived yet. Building on core math concepts through the incorporation of more real-world applications and further linking these concepts to statistics, computer science, and other STEM disciplines will help broaden the perspective of potential math majors, and better prepare them for the rest of college and their subsequent careers. This will not only create more well-rounded students, but will also strengthen math’s relationship with the other disciplines in the real world.

]]>When I started teaching, I wanted to be the very best teacher. Not just “the best teacher I could be”, but the *very* best teacher, the one students would tell their friends about and remember fondly years later, the kind of teacher they might imagine being the hero in a movie. I don’t know what your movie hero teacher looks like, but mine is beloved by all the students (more Robin Williams than John Houseman). So naturally, I wanted all the students to like me. I also wanted them to share my love of mathematics, and see it as a joyful endeavor, not just a requirement to be checked off. As a result, I started including more humor in my classes. What I eventually realized, and had to confront, was that at least some of what I was doing was more about making me look like that movie hero teacher, or about making the class fun, than about helping my students learn mathematics.

My first years of teaching, I would prepare for each class by writing notes of just about every word I would write on the board. (Like a low-tech Powerpoint presentation.) However, this greatly facilitated the sort of class where, as the saying goes, the ideas travel from my notes to my students’ notes, without having to pass through the brains of any of us. Because the other trait I imagined in my movie hero teacher was making the students truly understand (and not just memorize) the material, over time I brought in activities to encourage more active learning and interaction, which also made my classes less tightly-scripted.

I also started loosening up and allowing more of my personality and sense of humor to show, for instance slipping in more clever cultural references or ironic asides. This is part of who I am, how I communicate outside the classroom, even when discussing serious mathematics with colleagues. Sometimes it’s just hard to avoid. One of the small ways I have of making class more interactive is to ask students to help me with a proof or equation. After watching countless episodes of Blue’s Clues when my son was little, I find it almost impossible to do this without saying “You *will* help me, won’t you?“, just the way Steve, the show’s host, said it.

If I thought of anything amusing related to what we were discussing in class, I would share it. The payoff for this sort of thing is immediate, in the smiling or laughing faces of students. I could justify it by noting it made class more enjoyable, and maybe helped students remember ideas better. And it made me feel more like the movie hero teacher.

Now, shortly after I started teaching, I served on the jury for a trial. The lead attorney for one side had very much the attitude in court I was trying to cultivate in the classroom. He seemed to want to be friendly with us on the jury, and, while I don’t think he introduced any actual humor, he was certainly very relaxed and smiled a lot. The attorney for the other side was more down to business. I definitely liked the first attorney more, but I found myself sometimes a little irritated at him for being less serious.

I would occasionally remember this trial as I grew more comfortable injecting humor in the classroom. And I eventually started questioning my motives. Was I doing this because it helped students learn mathematics, or because I wanted them to like me? Here’s the final note of cognitive dissonance that made me confront myself. I would tell students we didn’t have time in class for some things, such as a review for an exam. But if we don’t have time for a review, how do we have time for a joke?

And here’s another concern. The clever wordplay and cultural references that I love so much have a special risk when dealing with students from another culture, or who are still mastering the language. I am especially aware of this, being on the border with another country. (It is literally true that I can see Mexico from my office window.) A number of years ago, teaching Game Theory, the textbook referred to the two players of some game as “Norm” and “Cliff“. I asked my students who understood this reference (click on the links to see the answer);So what did I change? I’ve kept in my cheerful outlook and my sense of wonder at mathematics (for instance overreacting for dramatic effect when some calculation yields a surprising result). But for anything beyond that, I now have two criteria for including anything entertaining:

**Does it take away precious class time?**Class time together is one of our most limited resources, and so I want to reserve it for the most important things, the ones that cannot be done individually or outside of class.**Does it unnecessarily distract the class from important mathematics?**Does what I am thinking of*reinforce*the mathematics, or is it just funny? Very rarely, it is worthwhile to take a short mental break to give everyone a chance to catch their breath. But I also recall Gian-Carlo Rota’s observation that a valuable trait for doing mathematics is Sitzfleisch, the ability to sit and concentrate for long periods of time.

And, with some practice and careful attention, I have now pretty much trained myself to avoid anything that fails either of these tests. I can now think of a funny idea, and consciously choose to not share it. For instance, here is a joke I used to tell when the idea for uniqueness came up:

How do you catch a unique rabbit? You “neak” up on it.

How do you catch a tame rabbit? “Tame” way, u-nique up on it.

The problem is, students may remember this joke, and so may remember the word unique, but does it help them understand the idea, or remember how to show something is unique, or anything mathematical at all?

On the other hand, some jokes help make a point.

Two campers are in their tent in the woods when they hear a bear. The first camper starts putting on shoes, and the other camper says, “You don’t have time for that, shoes aren’t going to help you outrun a bear.” The first camper replies, “I don’t have to outrun the bear, I only have to outrun

you.”

I tell this in Calculus I as we are starting global optimization, to make the point that sometimes it’s not enough to just have a very good solution or almost the best solution, but you have to have the best solution, second to none. Here, the joke may help to drive home this larger point, which some students may overlook in the thicket of algebra and derivatives that will shortly arise when we get to practice problems.

One final anecdote: In Linear Algebra, I was trying to make the point that even though we can’t put all matrices into diagonal form, we *can* at least put all matrices into Jordan canonical form (of which diagonal form is a more special case). I reminded students that not all matrices are diagonalizable, and wrote on the board, “You can’t always get what you want.” I realized only as I finished writing it that, without intending to, I was quoting the Rolling Stones. But in this case, it worked out perfectly because the next line (as realized by some students, who then practically sang it to everyone else) is “But if you try sometime, you just might find you get what you need,” which reinforced exactly the point I was about to make about Jordan canonical form.

And, come to think of it, this is a pretty good summary of where I am now. There will always be a part of me that still *wants* the adoration that comes from being the movie hero teacher. But what I *need*, what my students need, is for them to learn mathematics. And, as the song says, if I “try sometime,” I can give my students that opportunity while still sharing my enthusiasm for mathematics.

Oh, and if you want to hear the joke about the mathematician in the balloon, you can stay after class.

]]>Last year, I began offering an online Number Theory and Cryptography course for gifted high school students through Georgia Tech. Fourteen high school seniors from metro Atlanta took the course in Fall 2014, and overall I would say it was a big success. We will be offering the course again in Fall 2015 and are expecting roughly double the number of students. After describing the structure of the course, I will relate some of my experiences and describe some of the things I learned along the way. I hope this article stimulates others to think outside the box about using technology in education without necessarily following the standard “MOOC” model.

I was motivated to create this course because I benefited tremendously from a Saturday course on Linear Algebra and Differential Equations when I was a senior in high school — this was one of the key formative experiences which eventually turned me into a successful mathematician. However, I know that most public high school students don’t have that kind of opportunity available through their school systems, and I wanted to see if I could use modern technology to deliver an experience comparable to the one I had without requiring students to show up in person once a week.

Students qualified for the course by successfully completing Georgia Tech’s Distance Calculus program (which covers integral calculus, linear algebra, and multivariable calculus) by the end of their junior year in high school. My course covered basic number theory (e.g., modular arithmetic, primality testing, discrete logarithms, and quadratic reciprocity) and applications to cryptography (e.g., Diffie-Hellman key exchange, RSA, and El Gamal), plus some fun applications to topics like calendar calculations, music theory, and card tricks. A detailed syllabus can be found here: http://www.math2803.gatech.edu/wp-content/uploads/2014Math2803Syllabus.2.pdf

Students also learned to write proofs and to program in SAGE, and many of them learned LaTeX as well (all homework solutions had to be typed). Final projects included a calendar calculation quizzer for Android phones, a Power Point presentation on Furstenberg’s “topological” proof of the infinitude of primes, and a video on elliptic curve cryptography.

The course followed an asynchronous distance learning model, meaning that students watched videos on their own time rather than participating remotely in a live classroom. They also had weekly homework and reading assignments (from either the course textbook or supplemental handouts). There were three hour-long midterm exams, proctored by teachers at the students’ high schools, plus a final project. Final projects were presented at an end-of-the-semester, day-long “mini-conference”, in which the students and I got to meet one another in person. Students received Georgia Tech credit for the course, and their tuition was fully paid through a state-funded financial aid program (ACCEL). I held weekly video office hours, and there were extensive discussions on the course Piazza page. (Piazza is an integrated online discussion platform designed specifically for academic courses.) The videos, homework assignments, syllabus, and other course materials were organized through a password-protected WordPress site hosted by Georgia Tech.

Most of the videos for the course were filmed during a “pilot” offering of a similar course on campus in Fall 2013. All lectures for that course were videotaped, and my postdoc Greg Mayer helped me edit all the video footage. We broke up the material into roughly 5 minute chunks, uploaded the resulting videos to Videopress, and added descriptive captions below each video. The videos were then grouped into 14 weekly lessons, each (except for the last one) with an accompanying reading assignment and homework assignment. Sometimes it was necessary to add subtitles, for example when a student asked a question which wasn’t audible, or when I referred to something from a previous lecture which got edited out. Click here or here for some sample videos from the course.

The Piazza discussions were productive, as students asked a lot of questions and often answered each other’s questions before I had a chance to respond myself, which is great! Video office hours were less successful: usually only a few students would join in and they typically did not want to turn their cameras on, so these became chat sessions which could have been conducted just as efficiently via Piazza. I tried having a few discussion activities during the semester, with students working together in small groups via Piazza, but somehow this did not end up being one of the more effective aspects of the course. I will probably eliminate that aspect of the course next semester. On the other hand, there were some Piazza-facilitated interactive homework assignments which worked very well, such as an activity where students posted RSA public keys to Piazza and then encrypted and decrypted messages to one another. I plan to incorporate more such activities the next time around. Students got extra credit for learning how to calculate (in their heads) the day of the week given a date, and I tested them by video chat (using Adobe Connect). This worked reasonably well — much better than the usual office hours — though I don’t know how this approach would scale.

Overall I think the above format worked quite well, and I’m planning to use the same basic structure this coming Fall when I offer the course for a second time. We conducted extensive surveys before, during, and after the course and students seemed to enjoy the course and reported having learned a great deal. Georgia Tech’s Distance Calculus program is conducted in a synchronous format, meaning that the lectures are broadcast live, and since all the students in my course had previously taken part in that program, I thought it would be interesting to ask them which format (synchronous or asynchronous) they preferred. Many students preferred the asynchronous format, and none expressed a strong preference for the synchronous format. Most students reported spending an average of 6-9 hours per week on the course. A typical comment was “I didn’t expect to spend so much time on the assignments, but they were interesting, so it was worth it.”

I did encounter some problems. At the start of the course there was only one female student enrolled, and she dropped just before the first midterm exam. That left me with 14 men and no women. I could speculate on the reasons for this, but the bottom line is that I want to see a better gender balance in the future, and I will be discussing how to achieve this with our admissions office.

Two other issues that I continue to wrestle with are the sustainability and scalability of this course model. To help make the course more sustainable, in Fall 2015 I will teach the course both as a high school distance learning course and as a flipped course for undergraduates in the Georgia Tech Honors Program. (By flipped, I mean that students will watch videos and read content before class, and then I will meet with them face-to-face once a week in a workshop-style format; the high school and Honors Program sections will interact through Piazza, but not face-to-face except for the end of the semester mini-conference, to which both sections will be invited.) In theory I would like to eventually offer the course to students outside the state of Georgia, but that would probably involve more grading than I can do by myself, and it also raises currently unresolved issues regarding tuition, proctoring of exams, etc. So the ultimate level of scalability is still unknown.

I received a lot of assistance with this course, including help from Greg Mayer, who built the WordPress site in addition to editing the videos; from Nick Culpepper, an undergraduate student who graded all the homework assignments; from Georgia Tech’s Professional Education department, which handled proctoring and mailing of exams as well as the videotaping of the pilot course; and from the Admissions Office, which handled accreditation, registration, and tuition payment. I also had help from Georgia Tech’s School of Mathematics and College of Sciences, as well as from our CEISMC (Center for Education Integrating Science, Mathematics, and Computing) program. I would not recommend embarking on a project like this without a helpful and professional team like I had — they really made the whole experience quite enjoyable and (relatively) painless.

Note: This article is also posted at https://mattbakerblog.wordpress.com/2015/07/01/number-theory-and-cryptography-a-distance-learning-course-for-high-school-students/.

]]>**What’s the Directed Reading Program?**

“The Directed Reading Program (DRP) pairs undergraduates with mathematics graduate student mentors for semester-long independent study projects.”

This mission statement isn’t mine — it was the consensus of a group of graduate students at the University of Chicago in 2003. Since then, programs with this mission have been started at Rutgers, UConn, Maryland, MIT, UT-Austin, and UC-Berkeley. I was an undergraduate participant in the program at Chicago, and I founded the Maryland DRP in 2011. Since then our committee has overseen more than a hundred projects — freshmen through seniors, projects on areas as diverse as logic and finance, with student talks ranging from how to multiply complex numbers to a showcase of original research on nonlinear dimension reduction.

Here’s a bit more detail about how we run the program at Maryland. In the first week of the semester, graduate students in mathematics (including applied mathematics and statistics) and undergraduates (mostly math majors, but not all) submit a form telling us what they’re interested in doing. This doesn’t have to be a specific project. Sample undergraduate interests: “I enjoyed real analysis,” “Machine learning with real-world data,” “I talked about differential geometry with a professor in the department,” “My friend told me about topology and I think it sounds really cool.” Graduate interests tend to be more focused, often suggesting specific projects and the background a student needs for it. The DRP committee (a handful of graduate students) does our best to pair the students with mentors. After the first meeting, the mentor drafts a prospectus outlining the specific goals of the project. The undergraduate is expected to do about four hours of independent reading per week, to meet for a *conversation* with their mentor for an hour a week, and to give a 12-minute talk to other participants in the program at the end of the semester.

**That sounds neat, but why should I, a graduate student, start a DRP at my university? What are the benefits?**

This is a serious question — it’s critical that a school’s DRP be started and run by its graduate students. While our department gives us some money for pizza, mentors don’t get paid and students don’t get course credit.

One perspective is that this program can be a big deal for the undergraduates who participate, so it’s something that should be done. “The DRP program has given me a feel for what research is all about. Since I’ve gotten to explore my interests more, I’ve started to do research with a professor and am definitely planning on going to graduate school in mathematics.” Said differently, the DRP allows structured access to the cultural norms of the graduate community. Graduate students share anecdotes and legends of proofs gone right and wrong, we talk about professors and their weird teaching habits, and (shockingly!) we’re real people. Meeting one-on-one with an undergraduate allows us to bring some of our reality to them, and that allows them to see graduate school as a potential reality.

The other side of this is not why mentors should participate (graduate students should be doing lots of things!) but why people actually do sign up to mentor. “Because it’s fun!” “I taught a college algebra course this semester; it was great to spend some time working with a math major.” “He was excited about the stuff that I find interesting.”

Here’s a longer story, from a project I mentored — my student applied to the program to find out “how mathematicians think.” We started talking about intro analysis, and after a bit I suggested a pretty classic epsilon-delta problem — show that \(x^3 + ax^2 + bx + c\) tends to infinity as \(x\) gets large. The usual proof divides out the \(x^3\), then bounds the rest above \(0\). This method is so standard that I wasn’t expecting anything else. In particular, I definitely wasn’t expecting induction on the degree of the polynomial. If you take the derivative enough times, you get a linear function which goes to infinity, which means that eventually it’s bigger than \(1\). Now apply the Mean Value Theorem. (Awesome, right?) This proof won’t get either of us published, and I doubt that we’re the first ones to do it this way. But it’s ours to me, and the ownership of knowledge — knowing what I know, and knowing that there’s so much more — is a large part of my identity as a mathematician.

**Okay, I want to start a DRP. Do you have any logistical advice?**

- Having a diverse committee of graduate students is important. Because the success of a project is about mutual subject interest and trust between the student and mentor, pairings are much more likely to work when there’s someone on the committee who knows the mathematical content (and the personalities!) of each mentor who applies. Some of my friends who work in abstract algebra are happy to run a project on introductory number theory with a rising sophomore; others would prefer only to participate with a student who was ready to work through a chunk of a graduate-level text. (And that’s fine! We want both of these.) Maryland’s mathematics department in particular is massive, so it’s necessary for us to have people we can talk to about those same personalities of the graduate students in signal processing, or geometry, or applied statistics, or…
- Relatedly, decentralization works well. We used to have a big meeting at the start of the semester, but the one-on-one nature led to lots of timing conflicts. Things have gone much smoother since we’ve assigned committee liaisons for each pairing and let them schedule things individually.
- We try to be as inclusive as possible, but it’s tricky to reach people who aren’t math majors! We put up flyers in the math building and send out emails to the math department listserv each semester, but a lot of interest in the program comes from word-of-mouth.
- Projects can vary widely in scope, and that’s great. The important thing is that the student and mentor agree on a specific goal soon after their first meeting. Having a specific theorem/application/result in mind gives pairings a lot more direction and keeps students from feeling overwhelmed. This also helps with choosing an appropriate resource: for many students this will be a chapter or two from a textbook, but we’ve also had very successful projects that used an online open course.
- If you make the end-of-semester talks a priority, the talks will be
*really, really good*. Our policy now is to make sure that students have two full weeks to prepare their talks. Something I’ve noticed is that my students have often wanted to use some of this time as an opportunity to learn how to TeX. Sharing ideas is part of math; give them time to practice that!

**Is there anything else that would make it easier to start a DRP at my school?**

The Maryland DRP committee has made it a priority to share our resources — we want getting started to be as easy as possible. There are links on our website to all the forms that we use each semester. You’ll also find a list of all the talks that have been given since we started the program here (some of them even have slides!). If you have specific questions, you can also send me an email (sbalady at gmail dot com).

In my experience, the biggest obstacle to starting a DRP happened before I did anything — I spent a long time trying to convince my friends and professors that a program like this could work at our university. It was a lot easier just to do it! “The Directed Reading Program (DRP) pairs undergraduates with mathematics graduate student mentors for semester-long independent study projects.” Send out an email to the grad students in your department with that sentence and ask for volunteers; that’s really all it takes.

]]>My research focus is on undergraduate students’ solving of counting problems, and I have worked toward better understanding students’ combinatorial thinking. Counting problems provide excellent opportunities for students to engage in meaningful mathematical tasks and to experience tangible beynefits of being precise and meticulous in their work. In this post, I draw on my experience studying undergraduate students’ combinatorial reasoning to offer examples of “careful” work. There is likely little debate that it is important for students to be organized, precise, and careful as they engage in mathematical activities. Although some students turn in homework assignments that are detailed, organized, and well thought out, others pass over details or do not properly represent ideas. What makes some students (and not others) willing to invest time and effort in detailed and methodical work? How can we help students more amenable to being careful and precise? I believe that these are important questions to consider, and in this post I suggest moving toward emphasizing and characterizing this kind of behavior. In this post, I offer three contrasting examples of students’ solutions to counting problems, which highlight characteristics of careful and precise work.

When solving counting problems, students often simply take a guess at an answer, sometimes remembering (or misremembering) a formula, without being careful about generating a solution that makes sense and that can be justified. However, if students get in the habit of being more careful and methodical about identifying outcomes and carefully considering how to count, they can more easily avoid common counting errors.

Consider three students’ responses to a problem that states: “Fred, Jack, Penny, Sue, Bill, Kristi, and Martin all volunteered to serve on a class committee. The committee only needs 3 people. How many committees could be formed from the 7 volunteers?” This problem can be solved in a straightforward way by selecting three of the seven people to serve on the committee, yielding C(7,3) = 35. For students who are not familiar with binomial coefficients, a common response is to create an organized list that reflects a sum (seen in Student 3’s response below). A common incorrect solution would be to count arrangements (rather than sets) of people, which would be 7*6*5=210. This is incorrect because it counts arrangement of people within a committee, thus overcounting the total number of committees. Below, I compare and contrast three students’ written responses to this problem.

First, notice that Student 1’s response reflects the common incorrect answer, and we see that the student computed the product of 7*6*5. There is no attempt to check or verify the answer, nor does the student appear to consider smaller cases or to list potential outcomes. This response is not completely unreasonable, but it does not reflect particularly careful or precise work.

Student 2’s response provides evidence of an attempt at articulating outcomes and connecting those outcomes to a counting process. However, although the list seems organized in some ways, it is not constructed precisely enough to effectively yield a correct answer to the problem. The 7*3 =21 reflects the total number of listed outcomes, but it does not suggest a counting process that corresponds to the list. Student 2 is more organized than Student 1, but the approach still lacks care and precision necessary to answer the problem correctly.

In contrast to both of the other responses, Student 3’s work demonstrates a precise, systematic list of all of the outcomes. In this case, the careful mathematical work is seen in listing outcomes in a methodical and organized way. Specifically, the way in which the outcomes are listed actually reflects an overall structure (the 15+10+6+3+1) that helps to provide a convincing justification that no outcomes are missing or duplicated. Even more, the solution illuminates the recursive nature of the problem, and the level of detail the student included brings out more concepts and potentially more opportunity for generalization. For instance, one might be able to observe a pattern in the sums and generalize that the numbers of ways to choose three members from* n* is the sum of the first *n* – 2 triangular numbers. Student 3’s careful work thus affords opportunities for important and powerful mathematical connections.

What kinds of dispositions or experiences might lead a student to create a list like Student 3’s, and how can we develop those desirable traits? It might be the case that some students are predisposed to certain attitudes towards math, and this makes them more or less amenable to detailed work. Some students may have more stamina than others, and some may be more willing to engage in seemingly mundane activities to solve a problem. Regardless of existing predilections, I wonder if students can be convinced of the value of careful work. Perhaps by providing students at a variety of levels with opportunities in which they explicitly benefit from being precise, we could persuade students that such precision is worth their time.

Students should be given opportunities to see the value of engaging in careful and detailed practice – even if such work might initially seem unnecessary, boring, or overly elementary. Often, such additional effort – the patient, dedicated, and systematic approach that reflects a commitment to being careful and precise in one’s work – will pay off. We should put a concerted effort toward convincing students of this fact. Practically, this could look like giving students problems for which careful work reaps benefits – such as new mathematical insights discussed in Student 3’s work. Many problems have solutions that can be seen through a careful build-up of an argument through a series of smaller cases. Often, in order to develop these cases one needs to be organized, precise, and methodical not only in solving each case, but also in connecting smaller cases back to the original problem. In addition, many particular topics can lend themselves to careful and precise work. I have mentioned opportunities for precise work in counting problems, but I could see similar opportunities to highlight careful work in linear algebra, logic, and certainly more broadly in the development of proof.

The Common Core State Standards for mathematics (CCSSM) lists eight mathematical practices that students should adopt over time. Although the CCSSM present standards for students in K-12 mathematics classrooms, there is a need for precision at all levels of K-16 curriculum. One of these standards for mathematical practice is “Attend to Precision.” In the official description of this practice, the emphasis is primarily on communicating precisely, especially in using definitions, symbols, and units. I would argue that the kind of careful, detailed work I have described above is another potential way in which students can and should attend to precision. Indeed, mathematical problem solving involves a dialogue with oneself, and the written work on the page can be viewed as communication of ideas from the solver “back to” him or herself. From this perspective, work that is organized, carefully done, and precise could help formulate and solidify ideas for problem solvers, facilitating critical reasoning and successful problem solving.

]]>It has been one year since *On Teaching and Learning Mathematics* launched, so it seems an appropriate time for reflection. As I re-read the 36 articles we have published over the past twelve months, five prominent themes emerged that I will discuss below: teaching practices; bridges between K-12 and postsecondary education; expanding visions of mathematics education; the voices of students; and research, communication, and policy. If you have not had a chance to read all of our articles during the past twelve months, or if you have done so and would like to revisit them from a new perspective, this is my guide to the first year of our blog.

**Teaching Practices***.* One major theme of our blog over the past year has been the importance of interactions between students and teachers, particularly in classroom settings. From reflections on experimenting with varied pedagogical methods, to descriptions of interesting activities for students, to consideration of the role of broad student learning outcomes, the following articles provide many ideas that teachers can use to create quality interactions and engagement with students.

- Jerry Dwyer, Transformation of a Math Professor’s Teaching
- Benjamin Braun, Teaching Practices Between and Beyond All Lecture and All Student Discovery
- Art Duval, A Call for More Context
- Ryota Matsuura, The Hungarian Approach and How It Fits the American Educational Landscape
- Priscilla Bremser, Teaching Mathematics Through Immersion
- Oscar Fernandez, Helping All Students Experience the Magic of Mathematics
- Benjamin Braun, Assessment in Postsecondary Mathematics Courses
- Elise Lockwood, Reading Articles in Mathematics Education — It’s Not Just for Prospective Teachers!
- Janet Barnett, Dominic Klyve, Jerry Lodder, Daniel Otero, Nicolas Scoville, and Diana White, Using Primary Source Projects to Teach Mathematics
- Priscilla Bremser, Taming the Coverage Beast
- Benjamin Braun, Famous Unsolved Math Problems as Homework

**Bridges Between K-12 and Postsecondary Education**. What happens in K-12 education is important to postsecondary mathematics teachers for several reasons. Since postsecondary students are products of the K-12 system, what happens at the K-12 level has a clear impact on postsecondary mathematics education. Postsecondary mathematics educators have influence on the K-12 system, since K-12 teachers receive specialized mathematics content instruction in postsecondary classes. Further, core mathematical ideas and concepts transcend the K-12 to postsecondary divide, providing rich ground for making mathematical connections at all levels. The following articles address aspects of these issues and more.

- Art Duval, On Being a Friendly Mathematician
- Diana White, The Role of Mathematics Departments in the Mathematical Preparation of Elementary Teachers
- Sybilla Beckmann and Andrew Izsák, Why is Slope Hard to Teach?
- Dick Stanley, Proportionality Confusion
- Art Duval, One Reason Fractions (and Many Other Topics) Are Hard: Equivalence Relations Up and Down the Mathematics Curriculum
- Hung-Hsi Wu, The Mathematical Education of Teachers Part I: What is Textbook School Mathematics?
- Hung-Hsi Wu, The Mathematical Education of Teachers Part II: What are We Doing About Textbook School Mathematics?
- Elise Lockwood and Eric Weber, Some Thoughts on the Teaching and Learning of Mathematical Practices

**Expanding Visions of Mathematics Education**. Many of our articles emphasize the need to expand our vision of what it means to teach, learn, and use mathematics. Some of the following articles explore ways in which students’ and professors’ expectations of each other and themselves affect mathematical teaching and learning, while others meditate on the purpose and utility of mathematical knowledge and learning.

- Elise Lockwood, Striking the Balance Between Examples and Proof
- Keith Weber, Mathematics Professors and Mathematics Majors’ Expectations of Lectures in Advanced Mathematics
- Audrey St. John, The Power of Undergraduate Researchers
- Carl Lee, The Place of Mathematics and the Mathematics of Place
- William Yslas Vélez, Mathematics Instruction, an Enthusiastic Activity
- Priscilla Bremser, The Liberal Art of Mathematics
- Reinhard Laubenbacher, You Can Do Anything With a Math Degree

**The Voices of Students**. An often-neglected aspect of mathematics education is the reality of the experiences of our students. The following articles were written by students about their experiences, good and bad, providing teachers with a window into the world of mathematical learners.

- Morgan Mattingly, Transformation of a Math Student’s Learning
- Sarah Andrews, Justin Crum, and Taryn Laird, We Did the Math! Student Perspectives on Inquiry-Based Learning
- Sarah Blackwell, Rose Kaplan-Kelly, and Lilly Webster, Community, Professional Advice, and Exposure to New Ideas at the Carleton Summer Mathematics Program
- A.K. Whitney, In Math as in Dance, Don’t Miss a Step, or Else You May Fall

**Research, Communication, and Policy**. The final theme that stood out to me in our articles is the growing importance of postsecondary mathematics education research, communication among participants in the mathematics education community, and the impact of policies affecting higher education. Postsecondary mathematics education is in a state of transformation, and the following articles give a sense of how this transformation is manifesting itself with regard to how we understand learning, how students experience mathematics courses, and expectations for mathematics departments at institutions of higher education.

- Estrella Johnson, Karen Keene, and Christy Andrews-Larson, Inquiry-Oriented Instruction: What It Is and How We Are Trying to Help
- Priscilla Bremser, Do Mathematicians Need New Journals About Education?
- Diana White, The First Two Years of College Mathematics: Reflections and Highlights from the CBMS Forum
- Benjamin Braun, “The Time Has Come”: Highlights of the 2014 AMS Committee on Education Meeting
- Martha Siegel, Creating the 2015 CUPM Curriculum Guide
- Karen Saxe, Collective Action: Why the Future is Brighter for Undergraduate Teaching in the Mathematical Sciences

To conclude this article, I would like to thank the other members of the editorial board and our many invited contributors for the time and effort they have invested in this blog. All of our articles go through an editorial review process, meaning that every article we publish goes through multiple feedback/revision cycles. I have greatly enjoyed reading these contributions and watching our readership grow. I am looking forward to seeing what the future will bring!

]]>A remarkable event took place a few weeks ago at the Alexandria, Virginia headquarters of the American Statistical Association. Leaders from five professional associations whose missions include teaching in the mathematical sciences came together to guide future progress to incrementally improve education in our fields. It is the first time that all five — the American Mathematical Association of Two-Year Colleges (AMATYC), the American Mathematical Society (AMS), the American Statistical Association (ASA), the Mathematical Association of America (MAA), and the Society of Industrial and Applied Mathematics (SIAM) — are working together. Our focus is the collection of credit-bearing mathematics courses a student might take in the first two years of college. We examine the undergraduate program using a wide-angle lens, inclusive of modeling, statistics, and computational mathematics as well as applications in the broader mathematically based sciences.

**Why now?**

Each year approximately 50 percent of students fail to pass college algebra with a grade of `C’ or better.[1] Failure rates under traditional lecturing are 55 percent higher than the rates observed under active learning.[2] Undergraduate education in the mathematical sciences is in crisis in the United States. This crisis will affect all mathematical scientists at post-secondary institutions, regardless of each individual’s level of interest in education.

The crisis in mathematical sciences education is well documented in high-profile reports such as the U.S. government’s PCAST report on STEM education and the National Academies’ report on The Mathematical Sciences in 2025. In response (or in some cases, in anticipation of) these reports, various mathematical science associations have on their own or in collaboration released reports such as

- Committee on the Undergraduate Program in Mathematics Curriculum Guide
__[3]__ - Modeling Across the Curriculum
- Undergraduate Degree Programs in Applied Mathematics
- Partner Discipline Recommendations for Introductory College Mathematics
- Beyond Crossroads
- Guidelines for Undergraduate Programs in Statistical Science
- Guidelines for Assessment and Instruction in Statistics Education

There have been, and continue to be, many successful initiatives aimed at addressing the challenges identified. However, we believe it is time for *collective* action. We can no longer say, “I don’t teach those classes,” or “I don’t teach those students,” because students are now more mobile than ever, transitioning between multiple postsecondary institutions. For example, the National Student Clearinghouse Research Center’s *Two-Year Contributions to Four-Year Degrees* report found that 46 percent of all students who completed a degree at a four-year institution in 2013-14 had been enrolled at a two-year institution at some point in the previous 10 years. Research on “collective impact” suggests that, in achieving significant and lasting change in any area, a coordinated effort supported by major players from all existing sectors is more effective than an array of new initiatives and organizations.[4]

To maintain a viable workforce for our country, to continue the expansion of scientific knowledge, and to remain relevant, we must update our curricula, make current our pedagogical methods, connect more strongly to other disciplines, and perhaps even evolve the culture of our own discipline. Many in our own community predict that if we do not achieve large-scale improvement in undergraduate education on our own, then markets, governments, or other structures will force change upon all of us. We believe it is better to have agency in making the necessary changes.

Ben Braun’s recent blog post, which gives an account of the October 2014 AMS Committee on Education (CoE), states that “the most prominent theme of the meeting was the critical role of collaboration and cooperation at many levels: among department members, at the institutional level among departments and administrative units, among professional societies with common missions, and at the national level to ‘scale up’ successful models for effective teaching.” It is very good news indeed that important stakeholders are involved. A group of prominent mathematicians has come together to form Transforming Post-Secondary Education (TPSE Math) and they have recently published their first report. The umbrella organization for professional associations in the mathematical sciences, the Conference Board of the Mathematical Sciences (CBMS) held its forum on the first two years of college math, and is discussed by Diana White in her November 2014 blog post. Common Vision brings together the five professional associations whose missions include teaching in the mathematical sciences; it is our view that bringing association leadership together to work on undergraduate education is critical for lasting change.

Collective action to improve teaching and education in the mathematical sciences appears to be gaining traction.

**Who was at the workshop?**

The Common Vision 2025 project encourages action by highlighting existing efforts and draws on the collective wisdom of a diverse group of stakeholders to articulate a shared vision for modernizing the undergraduate mathematics program. We embrace the diversity of experience of our members.

Workshop participants included AMS President Robert Bryant, as well as several current and past presidents of all five associations. Participants also included faculty members from large departments at research universities; a statistician working at Google; a mathematician working at an HBCU; a vice president from the New York Hall of Science; faculty members from liberal arts colleges; faculty members from large comprehensive universities; the Executive Vice President of the APLU; a chemist working at the American Chemical Society; and an Achieving the Dream project director.

**What can you do? **

In reaching out to the membership of the five associations (including through this blog post) we hope to galvanize our colleagues and spur on a grassroots effort to improve education in the mathematical sciences.

Read the reports listed above. Read the Common Vision report, which will appear later this year and identifies common themes found in the above reports in order to provide a snapshot of the current thinking about undergraduate mathematics and statistics programs. Our report will also include a list of project ideas generated at our workshop. For example, you might identify a part of your curriculum that you would like to change in some way (like the calculus sequence, or the collection of upper level analysis courses, or the courses that do not require calculus and are intended for non-majors), and organize a meeting this summer with your colleagues about it; in advance, start a Google document where you can share ideas. Small changes, including more care and intention about our curriculum, can help our students have a better classroom experience. The activities are ones where we deem “small wins” are realistic, and are aimed at updating the mathematical sciences curriculum, updating pedagogical methods to align with best practices, and changing the culture of our discipline.

Please, do something. __Do__ something. Do __something.__

The Common Vision website: http://www.maa.org/common-vision

[1] Mathematical Association of America (2012). *Partner Discipline Recommendations for Introductory College Mathematics and the Implications for College Algebra*. Retrieved from Mathematical Association of America website: www.maa.org/sites/default/files/pdf/CUPM/crafty/introreport.pdf.

[2] Freeman, S, Eddy, S., McDonough, M., Smith, M., Okoroafor, N., Jordt, H., and Wenderoth, M.P., *Active learning increases student performance in science, engineering, and mathematics. *Proceedings of the National Academy of Sciences. Vol. 111. No. 23. June 10, 2014.

[3] See Martha Siegel’s blog post.

[4] Kania, J. and Kramer, M. (2011). *Collective Impact*, Stanford Social Innovation Review, Winter 2011.

When you return to the classroom as an adult student, a big perk is that what seemed like an unreasonable demand back then from the instructor suddenly makes sense, because maturity means you’re better able to fit it into the bigger picture. For me, a longtime journalist who decided to retake high school math at a community college after decades of hating and fearing it, that demand was “show your work.” As a teen, I’d always sighed when the teacher marked me down for not showing how I’d worked out a problem on an exam or in the homework. Why was it necessary to take eight steps to show a triangle’s angles added up to 180? What a bore.

But 20 years later, going from pre-algebra to calculus, I finally understand why, and I credit dance.

Huh? Let me explain.

To get to the math building on my community college’s campus, I’d usually take a shortcut through the dance department. I’d walk down a long corridor lined with mirrored studios, and no matter what kind of music was blaring out the doors – salsa, tap, jazz – an instructor would always count out a beat before the students began.

“And a one, two, three, four, five, six, seven, eight!”

Hearing this every class day, I not only realized that numbers were everywhere, but also that learning how to solve a math problem was a lot like learning how to dance. In both, there’s choreography involved, going from step one to step two to step three. And, at least in the case of ballroom dances like the fox-trot or waltz or cha-cha, there is a strict order of operations.

You may be Please Excusing My Dancing Aunt Sally, not my Dear one (PEMDAS), but missing a step or doing it out of order will really mess up the end result. Or else, it will turn the dance into something completely different.

By thinking of math problems that way, I was better able to tolerate my instructors’ endless insistence that I show all my work, especially on tests. I finally appreciated that they needed to know I truly grasped the elements of the problem, and that I respected the strategy needed to solve it. True, I still find it tedious to prove in eight steps that a triangle’s angles add up to 180 degrees, but I now know it’s good practice for way more complicated proofs, where thoroughness is key. I also appreciate that precision is vital to math, and if eight steps is what it takes to be precise in a triangle proof, so be it.

That said, a major peeve of mine, especially as I got further from applied math and closer to pure, was when instructors, while solving a problem, would take a sudden leap. This might entail doing quick factoring in a polynomial, going from 6x +6 to 6(x + 1) without explaining why it was necessary, or assuming students had memorized an obscure trigonometric identity, then making the substitution in a long equation without mentioning it.

I realize these are very simple examples, but depending on where I was in my math education, to me this was the dance equivalent of doing a two-step, then suddenly getting spun and landing on my butt. It would always take me a moment to regroup, and by then, I’d been left behind, standing against the wall and watching as everyone else whirled by. At least in class, I could try and stop the instructor and ask him or her to explain. But I always felt guilty about this, since we never seemed to have enough time to really get into the material. That guilt was spurred by the fact that every professor I had, from pre-algebra on, complained about class time never being enough to really go into depth on anything, especially if students didn’t grasp the material right away. And yes, all of these instructors had office hours for those slower students, but I discovered those hours were just as chaotic as they were in class, only now students were cramped into a tiny office, craning their necks to see what the professor was writing in a notebook. But that’s another discussion for another day.

It was worse when such a leap happened in the solutions manual. For the record, I’ve never much cared for these manuals, preferring to puzzle things out on my own. But sometimes I would come across a problem I just couldn’t solve, where it was all a blur, and I couldn’t pick out one step from the next. Looking at the worked out solution was a way to slow things down and get a guide.

However, when that guide skipped a step without explanation, there was no lecture to interrupt, no office to stalk. I was usually able to fill in the blank, but the time I spent doing so always had a cost. Sometimes it was not being able to get to all the other problems I needed to practice before the next test, or, more important, it dented my still fragile math confidence, making me unsure when I had to perform. And that anxiety sometimes led to failure on exams because I couldn’t relax enough to solve harder problems without second-guessing myself. Then I would make silly arithmetical mistakes on the other problems because I was rushing to catch up.

Now, I know that my inner demons were never my instructor’s problem. But that didn’t stop me from asking a few math professionals I came across why they skip steps while taking students through a problem.

Not enough class time, one said.

Including every step gets very tedious, said another, and you can lose sight of the bigger picture.

It is the student’s job to fill in the blanks, and doing so is the best way to retain the material, said several, though at least one added that this method worked best in classes more advanced than calculus. It can really backfire before then.

And these are all excellent reasons. But they didn’t help that jarring feeling of being spun, of falling, of landing badly, that I experienced when I revisited math after 20 years.

I understand that in math, as in dance, you have to get up and dust yourself off. And I did. But far too many of us don’t, which is why so many of us give up. And giving up on math has far worse repercussions, not just individually, but for all of society, than not becoming proficient at the fox-trot. And unlike me, most self-proclaimed math haters never return to the classroom.

So I ask the instructors reading this to consider shifting their perspective as I did mine. I accepted how important it was not to skip steps, to respect the choreography, so that you could see that I understood what was going on. You may have the best of reasons, but when you skip steps without explaining why, people like me, unused to the elaborate choreography, will fall down. We’re still learning. Don’t assume we can see how you did that leap. And hopefully, we’ll soon be dancing as gracefully as you.

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One of my favorite assignments for students in undergraduate mathematics courses is to have them work on unsolved math problems. An unsolved math problem, also known to mathematicians as an “open” problem, is a problem that no one on earth knows how to solve. My favorite unsolved problems for students are simply stated ones that can be easily understood. In this post, I’ll share three such problems that I have used in my classes and discuss their impact on my students.

**Unsolved Problems**

*The Collatz Conjecture*. Given a positive integer \(n\), if it is odd then calculate \(3n+1\). If it is even, calculate \(n/2\). Repeat this process with the resulting value. For example, if you begin with \(1\), then you obtain the sequence \[ 1,4,2,1,4,2,1,4,2,1,\ldots \] which will repeat forever in this way. If you start with a \(5\), then you obtain the sequence \(5,16,8,4,2,1,\ldots\), and now find yourself in the previous case. The unsolved question about this process is: If you start from any positive integer, does this process always end by cycling through \(1,4,2,1,4,2,1,\ldots\)? Mathematicians believe that the answer is yes, though no one knows how to prove it. This conjecture is known as the Collatz Conjecture (among many other names), since it was first asked in 1937 by Lothar Collatz.

*The Erd*ő*s-Strauss Conjecture*. A fascinating question about unit fractions is the following: For every positive integer \(n\) greater than or equal to \(2\), can you write \(\frac{4}{n}\) as a sum of three positive unit fractions? For example, for \(n=3\), we can write \[\frac{4}{3}=\frac{1}{1}+\frac{1}{6}+\frac{1}{6} \, . \] For \(n=5\), we can write \[ \frac{4}{5}=\frac{1}{2}+\frac{1}{4}+\frac{1}{20} \] or \[\frac{4}{5}=\frac{1}{2}+\frac{1}{5}+\frac{1}{10} \, . \] In other words, if \(n\geq 2\) can you always solve the equation \[ \frac{4}{n}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\] using positive integers \(a\), \(b\), and \(c\)? Again, most mathematicians believe that the answer to this question is yes, but a proof remains elusive. This question was first asked by Paul Erdős and Ernst Strauss in 1948, hence its name, and mathematicians have been working hard on it ever since.

*Lagarias’s Elementary Version of the Riemann Hypothesis*. For a positive integer \(n\), let \(\sigma(n)\) denote the sum of the positive integers that divide \(n\). For example, \(\sigma(4)=1+2+4=7\), and \(\sigma(6)=1+2+3+6=12\). Let \(H_n\) denote the \(n\)-th harmonic number, i.e. \[ H_n=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n} \, .\] Our third unsolved problem is: Does the following inequality hold for all \(n\geq 1\)? \[ \sigma(n)\leq H_n+\ln(H_n)e^{H_n} \] In 2002, Jeffrey Lagarias proved that this problem is equivalent to the Riemann Hypothesis, a famous question about the complex roots of the Riemann zeta function. Because it is equivalent to the Riemann Hypothesis, if you successfully answer it, then the Clay Mathematics Foundation will reward you with $1,000,000. While the statement of this problem is more complicated than the previous two, it doesn’t involve anything beyond natural logs and exponentials at a precalculus level.

**Impact on Students**

I’ve used all three of these problems, along with various others, as the focus of in-class group work and as homework problems in undergraduate mathematics courses such as College Geometry, Problem Solving for Teachers, and History of Mathematics. An example of a homework assignment I give based on the Riemann Hypothesis problem can be found at this link. When I use these problems for in-class work, I will typically pose the problem to the students without telling them it is unsolved, and then reveal the full truth after they have been working for fifteen minutes or so. By doing this, the students get to experience the shift in perspective that comes when what appears to be a simple problem in arithmetic suddenly becomes a near-impossibility.

Without fail, my undergraduate students, most of whom are majors in math, math education, engineering, or one of the natural sciences, are surprised that they can understand the statement of an unsolved math problem. Most of them are also shocked that problems as seemingly simple as the Collatz Conjecture or the Erdős-Strauss Conjecture are unsolved — the ideas involved in the statements of these problems are at an elementary-school level!

I have found that having students work on unsolved problems gets them engaged in three ways that are otherwise very difficult to obtain.

*Students are forced to depart from the “answer-getting” mentality of mathematics.*In my experience, (most) students in K-12 and postsecondary mathematics courses believe that all math problems have known answers, and that teachers can find the answer to every problem. As long as students believe this story, it is hard to motivate them to develop quality mathematical practices, as opposed to doing the minimum necessary to get the “right answer” sufficiently often. However, if they are asked to work on an unsolved problem, knowing that it is unsolved, then students are forced to find other ways to define success in their mathematical work. While getting buy-in on this idea is occasionally an issue, most of the time the students are immediately interested in the idea of an unsolved problem, especially a simply-stated one. The discussion of how to define success in mathematical investigation usually prompts quality discussions in class about the authentic nature of mathematical work; students often haven’t reflected on the fact that professional mathematicians and scientists spend most of their time thinking about how to solve problems that no one knows how to solve.

*Students are forced to redefine success in learning as making sense and increasing depth of understanding*. The first of the mathematical practice standards in the Common Core, which have been discussed in previous blog posts by the author and by Elise Lockwood and Eric Weber, is that students should make sense of problems and persevere in solving them. When faced with an unsolved problem, sense-making and perseverance must take center stage. In courses heavily populated by preservice teachers, I’ve used open problems as in-class group work in which students work on a problem and monitor which of the practice standards they are using. Since neither the students nor I expect that they will solve the problem at hand, they are able to really relax and focus on the process of mathematical investigation, without feeling pressure to complete the problem. One could even go so far as to evaluate student work on unsolved problems using the common core practice standards, though typically I evaluate such work based on maturity of investigation and clarity of exposition.

*Students are able to work in a context in which failure is completely normal.*In my experience, undergraduates majoring in the mathematical sciences typically carry a large amount of guilt and self-doubt regarding their perceived mathematical failures, whether or not it is justified. From data collected by the recent MAA Calculus Study, it appears that this is particularly harmful for women studying mathematics. Because working on unsolved problems forces success to be redefined, it also provides an opportunity to discuss the definition of failure, and the pervasive normality of small mistakes in the day-to-day lives of mathematicians and scientists. I usually combine work on unsolved problems with reading assignments and classroom discussions regarding developments in educational and social psychology, such as Carol Dweck’s work on mindset, to help students develop a more reasonable set of expectations for their mathematical process.

One of the most interesting aspects of using unsolved problems in my classes has been to see how my students respond. I typically ask students to write a three-page reflective essay about their experience with the homework in the course, and almost all of the students talk about working on the open problems. Some of them describe feelings of relief and joy to have the opportunity to be as creative as they wish on a problem with no expectation of finding the right answer, while others describe feelings of frustration and immediate defeat in the face of a hopeless task. Either way, many students tell me that working on an unsolved problem is one of the noteworthy moments in the course. For this reason, as much as I enjoy witnessing mathematics develop and progress, I hope that some of my favorite problems remain tantalizingly unsolved for many years to come.

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