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.

]]>By the end of every workshop and conference session on Inquiry-Based Learning that I’ve attended, someone has raised a hand to ask about coverage. “Don’t you have to sacrifice coverage if you teach this way?” Of course coverage took center stage for many of my professional conversations long before I tested the IBL waters; it’s important. But an equally important question is this: What do we sacrifice when coverage dominates? It may well be conceptual understanding; it’s possible to cover more ground, albeit thinly, if we settle for procedural understanding instead. More than once I’ve settled for even less, delivering a quick lecture just so that my students will have “seen” a particular idea. How do we strike a balance between coverage and other considerations when we are so practiced at reducing a course description to a list of topics?

Strong arguments for striking that balance have been made elsewhere. For example, Stan Yoshinobu and Matthew Jones offer a close examination of the “price of coverage”. “Coverage versus depth” is a “false dichotomy,” they say; racing through material makes for a passive student experience, which affects student understanding of what it is to learn mathematics. “Implied messages are sent to students through classroom experiences,” and some of those messages may have unproductive consequences, including overreliance on mimicking the instructor and memorization, and significant difficulties with non-routine problems.

Is there, on the other hand, a price of demoting coverage? Does a more comprehensive view of student learning get in the way of content knowledge? Recent research done by Marina Kogan and Sandra Laursen, brought to my attention by Yoshinobu and David Bressoud, suggests that students don’t necessarily suffer, and may be helped, from a holistic approach. From the conclusion to the Kogan and Laursen paper:

College instructors using student-centered methods in the classroom are often called upon to provide evidence in support of the educational benefits of their approach—an irony, given that traditional lecture approaches have seldom undergone similar evidence-based scrutiny. Our study indicates that the benefits of active learning experiences may be lasting and significant for some student groups, with no harm done to others. Importantly, “covering” less material in inquiry-based sections had no negative effect on students’ later performance in the major. Evidence for increased persistence is seen among the high-achieving students whom many faculty members would most like to recruit and retain in their department.

Still, it’s often difficult to prevent concerns about coverage from hijacking day-to-day teaching practice, regardless of course format. Here are some approaches I am using to keep coverage in perspective.

**Regard conceptual understanding, mathematical writing and speaking, and other learning goals as integral parts of the “coverage” list, on an equal par with specific topics.** Yoshinobu points out that we have a “systemic” issue, in that our institutions define coverage as no more than the list of topics. Hence I have to make a conscious effort, in planning each course, to weave all of the goals together, and to recognize that procedural skills won’t last without conceptual understanding, which in turn won’t happen if students don’t routinely speak and write mathematics.

**Include learning objectives, not just a topics list, on the syllabus. ** Whether or not all of my students read the syllabus, it’s my way of formalizing my intentions and expectations. It’s also an invitation to consider the course in its entirety. This is especially important in mathematics, where students don’t understand many of the terms in a catalog description until after they’ve taken the course.

**Have conversations with students, early and often, about the learning goals for the course.** On the first day of linear algebra this semester, I devoted the entire hour to a class activity adapted from a model offered by Dana Ernst. The students’ responses to “What are the goals of a liberal arts education?” included “critical thinking” and “to experience the freedom to explore.” To “What can you reasonably expect to remember from your courses in 20 years?” I heard, “NOT details or the stuff you’re tested on,” but rather “how to figure out what’s relevant.” My own students understand the big picture; surely I can keep it in mind!

Halfway through the term, I had my students read this blog post from Ben Orlin and then fill out a survey online. I asked: to what extent are you practicing in the Church of Learning, as opposed to the Church of the Right Answer? Once again, the students reinforced my choices. Many of them also noted that their pre-college experiences, especially Advanced Placement Calculus, leaned heavily toward the Right Answer doctrine. In at least some cases, I’m working against students’ most recent experience of mathematics learning, so I need to be persistently transparent.

**Gather data frequently on student understanding**. Formative assessment isn’t just for elementary school teachers. I’m fortunate to teach small classes, so I can learn a lot just from classroom conversations. In an earlier post, I explained how recent research on learning has influenced my teaching. If I hear someone struggling to use “linearly independent” accurately during small group work, I can offer corrective feedback immediately. My students often show their work using a document projector. Anonymous surveys are useful as well; it only takes a few minutes for students to write down what’s puzzling them at the moment. I’ve never used clickers, but I’m intrigued by Eric Mazur’s methods. Most importantly, I try to design homework assignments that ask for deeper understanding. (It takes several weeks to convince students that homework is for formative, not summative, assessment, and that the graders’ job is to give constructive feedback.)

**Bring student graders and teaching assistants in on the plan**. I handpicked my graders this term, and made it clear that I want homework solutions to be clear and well-written, not just correct. They know that I’ve encouraged the students to show their attempts and partial solutions to more challenging problems. They let me know what misconceptions they see. The student tutors are also aware of my intentions.

It may be that I am especially sensitive to questions about coverage because my semester includes only twelve weeks of classes. My department colleagues and I agree that this poses a particularly vexing challenge in multivariable calculus. Getting to Green’s Theorem is challenging enough, and a thorough treatment of Stokes’ Theorem, which would add coherence to the entire semester, seems a worthy goal. Yet even here, I remind myself, what’s important is not only what I cover; it’s also what the students can retain.

]]>Making fundamental changes to the way you teach is a difficult task. However, with a growing number of students leaving STEM majors, instructors’ dissatisfaction with student learning outcomes, and research indicating positive avenues for improving undergraduate mathematics instruction, some instructors are ready and eager to try something new. In this post, we describe some promising research-based curricular materials, briefly identify specific challenges associated with implementing these materials, and describe a recently funded NSF project aimed at addressing those challenges.

Teaching Inquiry-Oriented Mathematics: Establishing Supports (TIMES) is an NSF-funded project (NFS Awards: #143195, #1431641, #1431393) designed to study how we can support undergraduate instructors as they implement changes in their instruction. A pilot is currently being conducted with a small group of instructors. In the next two years, approximately 35 math instructors will be named TIMES fellows and will participate in the project as they change their teaching of differential equations, linear algebra, or abstract algebra. As project leaders, we will study how to best support these instructors, as well as how their instructional change affects student learning. More details about the project follow later in this blog post.

**Inquiry-Oriented Instruction
**

The curricula we utilize in the project are each examples of inquiry-oriented instructional materials. Inquiry-oriented instruction is a specific type of student-centered instruction. Not surprisingly, different communities characterize inquiry in slightly different ways. In the inquiry-oriented approach we describe here, we adopt Rasmussen and Kwon’s (2007) characterization of inquiry, which applies to both student activity and to instructor activity. In this approach, students learn new mathematics by: engaging in cognitively demanding tasks that prompt exploration of important mathematical relationships and concepts; engaging in mathematical discussions; developing and testing conjectures; and explaining and justifying their thinking. Student inquiry serves two primary functions: (1) it enables students to learn new mathematics through engagement in genuine exploration and argumentation, and (2) it serves to empower learners to see themselves as capable of reinventing important mathematical ideas.

The goal of instructor inquiry into student thinking goes beyond merely assessing student’s answers as correct or incorrect. Instead, instructor inquiry seeks to reveal students’ intuitive and informal ways of reasoning, especially those that can serve as building blocks for more formal ways of reasoning. In order to support students, instructors routinely inquire into how their students are thinking about the concepts and procedures being developed. As instructors inquire into students’ emerging ideas, they facilitate and support the growth of students’ self-generated mathematical ideas and representations toward more formal or conventional ones. The instructor’s role is to guide and direct the mathematical activity of the students as they work on tasks by listening to students and using their reasoning to support the development of new conceptions. Additionally, instructors provide connections between students’ informal reasoning and more formal mathematics.

With an inquiry-oriented instructional approach, instructors use mathematically rich task sequences, small group work, and whole class discussions in order to elicit student thinking, build on student thinking, develop a shared understanding, and introduce formal language and notation.

**Curricular Materials for Undergraduate Mathematics Education
**

The TIMES project is organized around three sets of post-calculus, research-based, inquiry-oriented curricular materials.

· Inquiry-Oriented Abstract Algebra (IOAA), developed by Sean Larsen under the NSF grant Teaching Abstract Algebra for Understanding (#0737299), http://www.web.pdx.edu/~slarsen/TAAFU/ (User:AMSBlog; Password:teacher). These materials are designed for an introductory group theory course and include units on groups and subgroups, isomorphisms, and quotient groups. Supplementary materials for rings/fields are available upon request.

· Inquiry-Oriented Linear Algebra (IOLA), developed by Megan Wawro, Michelle Zandieh, Chris Rasmussen, and colleagues under NSF grant numbers 0634074/0634099 and 1245673/1245796/1246083, http://iola.math.vt.edu (must request login & password). These materials are designed for an introductory linear algebra course and include four units on span, linear dependence and independence, transformations, and eigenvalues, eigenvectors, and change of basis. Tasks for determinants and systems are also available upon request.

· Inquiry-Oriented Differential Equations (IODE), developed by Chris Rasmussen and colleagues under NSF grant number 9875388, website coming soon. These materials are designed for a first course in differential equations and include the following topics: solving ODEs; numerical, analytic and graphical solution methods; solutions and spaces of solutions; linear systems; linearization; qualitative analysis of both ODEs and linear systems of ODEs; and structures of solution spaces.

For each of these three curricular innovations, the student materials have been developed through iterative stages of research and design supported by grants from the NSF. In the early stages of these respective projects, the developers carried out small-scale teaching experiments focused on uncovering students’ ways of reasoning and developing tasks that evoke and leverage productive ways of reasoning. Instructional tasks then went through additional cycles of implementing, testing, and refining over a series of whole class teaching experiments. In the last stages of research and design, instructors who were not involved in the development implemented the materials and provided feedback.

Over the course of the last 10+ years, these extensive and ongoing research projects have produced many results, including: instructional sequences comprised of rich problem-solving tasks, instructor support materials, research showing positive conceptual learning gains (e.g., Kwon, Rasmussen, & Allen, 2005; Larsen, Johnson, & Bartlo, 2013), insights into how students think about these concepts (e.g., Larsen, 2009; Wawro, 2014; Keene, 2007) and the identification of specific challenges that instructors face as they implemented these materials. Some of the difficulties experienced by instructors implementing the materials include: making sense of student thinking, planning for and leading productive whole class discussions, and building on students’ solution strategies and contributions (e.g., Johnson & Larsen, 2012; Speer & Wagner, 2009; Wagner, Speer, & Rossa, 2007).

**TIMES Project
**

The TIMES grant will allow us to better understand how to support instructors as they work to implement these three inquiry-oriented curricula materials. We have a three-pronged instructional support model, consisting of:

(1) Curricular support materials – These materials, created by the researchers who developed the three curricular innovations, include: student materials (e.g., task sequences, handouts, problem banks) and instructor support materials (e.g., learning goals and rationales for the tasks, examples of student work, implementation notes).

(2) Summer workshops – The summer workshops last 2-3 days and have three main goals, 1) building familiarity with the curricula materials, including an understanding of the learning trajectories of the lessons; and 2) developing an understanding of the intent of the curricula in particular and inquiry-oriented instruction in general.

(3) Online instructor work groups – The online instructor work groups have between 4 and 6 participants, each currently implementing the same curricular materials. Each group meets for one hour a week and works on selected lessons from the curricular materials. For each of the focal lessons, we discuss the mathematics and plan for implementation. Then, after instructors have taught the lesson, the group watches video clips of instruction with a focus on student thinking. The goal is to help instructors develop their ability to interpret and respond to student thinking in ways that support student learning. Every meeting also has time dedicated to address specific and immediate needs of the participants (e.g., difficulty with managing small group work, a particularly challenging task, strategies for getting students to share ideas).

Over the course of this three-year grant, we will offer these supports and investigate their impact. Our research will focus on the relationships and interactions among the supports, the instructors, and their instructional practices. In addition to assessing the impact of the support model, project data will be analyzed to identify aspects of the supports and instruction that have a positive impact on students’ learning.

We hope that this post provided a useful description of the inquiry-oriented instructional approach that can help instructors think about how they might (or already do) incorporate some of these ideas into their teaching. For instance, regardless of how you currently teach, really inquiring into your students’ thinking (not just their answers) can provide you with very valuable insights. We also hope that, after reading this post, you will be encouraged to see that some tangible, practical steps are being taken toward scaling up and supporting inquiry-oriented instruction.

If you are interested in learning more about the curricular materials or this project please visit http://times.math.vt.edu/. If you are interested in learning more about becoming a TIMES fellow, please contact Estrella Johnson (strej@vt.edu) for Abstract Algebra, Christy Andrews-Larsen (cjlarson@fsu.edu) for Linear Algebra, or Karen Keene (kakeene@ncsu.edu) for Differential Equations. We are the principal investigators on the project and would be glad to hear from you if you are interested in learning more.

**References**

Johnson, E. M. S., & Larsen, S. (2012). Teacher listening: The role of knowledge of content and students. Journal of Mathematical Behavior, 31, 117 – 129.

Keene, K. A. (2007). A characterization of dynamic reasoning: Reasoning with time as parameter. The Journal of Mathematical Behavior, 26(3), 230-246.

Kwon, O. N., Rasmussen, C., & Allen, K. (2005). Students’ retention of mathematical knowledge and skills in differential equations. School Science and Mathematics, 105(5), 227-239.

Larsen, S. (2009). Reinventing the concepts of group and isomorphism: The case of Jessica and Sandra. The Journal of Mathematical Behavior, 28(2), 119-137.

Larsen, S., Johnson, E., & Bartlo, J. (2013). Designing and scaling up an innovation in abstract algebra. The Journal of Mathematical Behavior.

Rasmussen, C., & Kwon, O. N. (2007). An inquiry-oriented approach to undergraduate mathematics. The Journal of Mathematical Behavior, 26(3), 189-194.

Wawro, M. (2014). Student reasoning about the invertible matrix theorem in linear algebra. ZDM, 46(3), 389-406.

Speer, N. M., & Wagner, J. F. (2009). Knowledge Needed by a Teacher to Provide Analytic Scaffolding During Undergraduate Mathematics Classroom Discussions. Journal for Research in Mathematics Education, 40(5), 530-562.

Wagner, J. F., Speer, N. M., & Rossa, B. (2007). Beyond mathematical content knowledge: A mathematician’s knowledge needed for teaching an inquiry-oriented differential equations course. The Journal of Mathematical Behavior,26(3), 247-266.

]]>When I first started teaching, I was mystified (and, frankly, at times panicked) at the thought of having undergraduates work with me on research. I realized this was part of the job, part of my institution’s mission, but I just couldn’t figure out how it would be effective. Sure, these students were bright, eager and motivated to learn, but how much could they contribute with such limited time? A typical research experience might be 8-10 weeks during the summer (full time) or 10 hours a week during a semester; best case, I might find a student who would work with me for a couple years in this way. I had just finished six years in grad school and still felt like I knew nothing. On top of that, my research is at the intersection of computer science and math with applications in the domains of engineering and biology – would I be able to find students with experience in even two of these fields? As it turns out, I would soon discover how powerful research with undergraduates can be, and I’d like to share some of the lessons I’ve learned over the years.

I remember thinking I should come up with a list of very specific problems, solvable with limited time and background, before trying to find students. Looking back, I think I was trying to mirror the familiar classroom experience, where a careful syllabus provides clear expectations to students with specified prerequisites. It turns out that specific problems, while useful in giving students an idea of the research area, almost never provide the direction we end up moving toward. I suppose I should have seen that coming, as research never does go as planned. It can’t be clean and predictable just because undergraduates are involved. I’ve learned to embrace the prospect of the unknown, instead looking for students with more broadly defined interests, such as computational biology or robotics.

Once I’ve found students, the most successful approach comes from guiding them along paths that suit their own passions and interests. Many times, they don’t really know these in advance, so I view the first part of my time with them as a chance to let them play with different types of problems. This may mean coming up with examples that fit a given set of combinatorial properties, reading and presenting a research paper on an algorithm we hope to generalize, or building modules of Mathematica code to explore properties of certain matrices. One summer, my students built little robots with microcontrollers and old VCR boxes; that activity resulted in one student determined to continue working on hardware and another determined to work only on software (a surprise to her). This “discovery period” can be truly transformative for some, and the reward of knowing that I helped a student find out a little more about herself is one of the main reasons I became a faculty member.

After identifying her interests, the student begins to get a glimpse of how research feels by facing the energizing and terrifying prospect of defining her own problems and pathway. I have the students pitch their own projects and timelines (which are always too ambitious) and work with them to create several milestones along the way. I usually let them start off on their overly optimistic timeline, but know they generally won’t make it past the first milestone. The students track their own progress by maintaining a website with blog updates on their work. This serves two purposes: (1) it helps me understand what they have done, what they understand and where they are stuck, and (2) it gives them something to reflect on at the end of the experience. Throughout this time, I am very conscious of each student’s confidence level. For some, the unfamiliarity of not having lectures, assignments and a textbook can cause them to doubt their own ability. Explicitly telling them that research is coupled with a feeling of the unknown and relating imposter syndrome stories of my own and of other researchers often gets them back on track. This is one of the things I enjoy the most, mentoring students who are excited to work on problems related to my research and helping them find the confidence to jump-start their own research careers. It is an amazing feeling when they tell me years later that it was that seed of a research experience which grow into their passion, whether it is pursuing a graduate degree with an NSF fellowship or becoming a teacher who will inspire new generations or working on cutting edge technology at an industry leader.

I used to worry that working with students on problems that interest them might be a distraction from my own research. I had, in earlier years, been asked by a local roboticist to help advise students on a project of his. I had no experience in robotics, but saw the excitement on the students’ faces. As the only faculty member positioned to co-advise, I knew that my saying “no” would crush their hopes. As I became more involved in their projects, helping to build a 3D printer from a kit, I became enamored with microcontrollers and the “maker” movement. At that point, it was just fun to build stuff and create an interactive project with a few lines of code; in my mind, it was completely decoupled from research. Then, two years ago, I began thinking about applying for an NSF grant, but was stumped as to what exciting research pathway I could propose. Serendipitously, a roboticist, whom I’d met through this robotics work, sent me a link to a TED talk with quadcopters cooperating to catch a ball in a net. A light bulb went off, and I saw a connection to the theoretical core of my research. I took a risk and proposed this robotics-based research program. To my delight and surprise, my proposal was funded!

This was not the only time a surprising connection came from working with a student. In fact, my first undergraduate researcher impacted my career in a way she may not even know. She sought me out one day as she was double-majoring in computer science and mathematics and had been told that my research straddled both. I felt completely unprepared as I had no list of specific problems; instead, I described my research on the fly, and my work in computational biology piqued her interest. She has since become a co-author with another undergraduate and two biochemists, but perhaps her most unexpected gift to me was a new collaboration, which I value deeply. As part of writing up her thesis, this student wanted to provide background on Lie groups, and she sought out the expertise of a mathematics professor. This professor saw a connection between the thesis work and the research area of another math professor. She encouraged us to start talking and thus began a collaboration for which I will be forever grateful.

As faculty at a research liberal arts institution, involving undergraduates in research is a core part of what we do. These budding researchers may not always be able to produce significant original contributions, but I can genuinely say my research path has been dramatically transformed for the better because of them. This is the biggest lesson I have learned: don’t underestimate the power of undergraduate researchers. They might directly contribute to your research, becoming co-authors on your next publication, or provide context and intuitions from things you’ve never thought about. And one day, those student interactions just might result in a connection that will transform who you are as a researcher. That connection could lead to a fantastic new collaborator or even a successful grant proposal. And, to top it all off, you get the amazing reward of knowing you played a small role in helping them in their own journey of discovering where to go next.

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