Here at the University of Colorado Denver, we’re starting our fourth week of classes. One of the classes that I’m teaching this semester is the history of mathematics. As part of an NSF-funded grant, Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS), I’m mentoring a graduate student in the use of primary sources projects in the classroom. This is helping to sustain my intentionality with regard to my preparation as well as my choice of instructional practices. In this role, I have been pondering both how to be a good mentor as well as how to keep working to learn and grow in my own teaching throughout the entirety of the semester.

This has led me to return to some of our past blog posts that I found particularly helpful to read or write, which I want to share. Below are links to some of these past blog entries which focused directly on some aspect of classroom teaching practices, and that I want to use throughout the next few months to keep my energy level up for my teaching. I hope you can find something here to energize you as well.

The first is a link to the editorial board’s six-part series on active learning that appeared in 2015:

https://blogs.ams.org/matheducation/category/active-learning-in-mathematics-series-2015/

This was followed up by an article in the Notices of the AMS from February 2017:

http://www.ams.org/publications/journals/notices/201702/rnoti-p124.pdf

This next entry by Steven Klee at Seattle University focuses on how to encourage increased student interactions during group work by having them work together at the board:

https://blogs.ams.org/matheducation/2017/09/18/do-we-get-to-work-at-the-board-today/

One of my all-time favorites, by Art Duval at the University of Texas at El Paso, focuses on if telling jokes and making class humorous is really beneficial to student learning, or if it unnecessarily takes away precious time that the instructor and students have together:

https://blogs.ams.org/matheducation/2015/07/10/dont-make-em-laugh/

And, finally, a post from Allison Henrich at Seattle University, reminding us of the wonderful value of mistakes in the learning process, and sharing ideas of how to help students be comfortable with making and discussing mistakes in the classroom:

https://blogs.ams.org/matheducation/2017/05/01/i-am-so-glad-you-made-that-mistake/

As you progress through your semester, I hope you find something in these various posts to keep you energized and growing in your own practice of teaching.

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It persistently rises to the surface of your memory – that afternoon when you fell in love with a person or a place or a mood … when you discovered some great truth about the world, when an indelible brand was seared into your heart, which is, of course, a finite space with limited room for searing.Arthur Phillips,

Prague

It was my senior year of high school. I had spent the first half of my day taking the AIME exam. At the end of the exam, there was one problem that really intrigued me. I couldn’t stop thinking about it! It lingered in the back of my mind through lunch and gym class. When I got to my history class, I had an idea to start looking at small examples: what if there were only two houses on the street? Or three? Or four? Then I had an “a-ha” moment, which let me see a recursive pattern and ultimately led to the solution of the problem.

The joy I experienced at solving this problem was profound, and it still stands out in my mind, almost 20 years later, as a significant moment in my mathematical journey. I had had this insight that was completely new (at least it was new to me), and led me to solve a problem that was unlike anything I had ever seen before. It was exciting! It didn’t count towards my grade anywhere, but that didn’t matter. I had discovered something new, and mathematics had left an indelible brand on my heart.

My goal in this article is to examine this experience more carefully, along with the experiences of other mathematicians and scientists, to try to understand the “a-ha” moments that can be so powerful for our students. To gather data, I asked a large group of people, including high schoolers, academics, and people in industry to reflect on the following question:

Tell me about one of the first times you ever experienced joy or excitement at solving (or not solving) a math problem. When did this happen? Do you remember the problem? What made this experience so memorable?

In what follows, I will reflect on general themes that surfaced in the responses I received in the hopes that they can help us more deeply reflect on our own teaching. I am grateful to my friends and colleagues who shared their stories. Each one was exciting and inspiring in its own way, and I regret that I was not able to include an excerpt from each of them. I would love to hear about *your* stories of joy and mathematical discovery in the comments section below.

Several people commented that their moments of inspiration came from venturing into the unknown of the mathematical landscape. Sara Billey (University of Washington) reflected on the joy of solving her first research problem:

One day after studying almost everything known about the problem, I decided to close the books, put away the previously published papers, and pull out a clean sheet of paper. I asked myself “What could I prove that was not written?” I wrote down a formula that combined a fact I knew from the literature with the problem I was trying to solve. I asked if that formula could also be true. I sat there for a while and the proof came to me. I wrote it up in my notebook, and declared that a successful day. About two weeks later, I showed this formula to another student, who got inspired to write down another, related formula. He came back a few days later and said he could prove the conjecture if a third formula was true. Well, I had the feeling I could prove the third formula by putting a bunch of things together. Sure enough, my rather intricate proof worked! It was a very exciting time, and I got to be a part of it because I forced myself to close the textbooks and ask myself a question beyond what was already written.

Matthias Beck (San Francisco State University) echoed these sentiments, writing:

I vividly remember the first original research problem I solved. I knew the literature well enough by that point that I was pretty sure that my theorem was novel, and that caused a certain sense of excitement: the thought that at this point in time nobody else had ever scribbled down what was written on my pieces of paper.

The power of making one’s mark on the mathematical landscape by discovering some fact that was previously unknown to the world is no doubt significant. The feeling of accomplishment that comes with a new research discovery has affected researchers at all levels, from undergraduate REU participants to established researchers.

On the other hand, this venture into the unknown need not be predicated on a research experience. A problem does not need to be new to the world in order for its solution to be meaningful; it just needs to be new to the student. Another respondent recalled her first memorable problem:

The problem was as follows: a pencil costs X, an eraser costs Y, and a pen costs Z. Can you buy these items in such a way that the total cost is M? The point of the solution was that X, Y, and Z were divisible by 7, but M was not. I was eleven at the time. It took me a few hours, but then it finally hit me how to solve it. I felt so excited when I finally got that “a-ha” moment.

In many cases students were moved because they had a sense of ownership of the problem and its solution. It is easy to feel a sense of ownership in research where we write papers with our names on them and other people refer to our results, but this same feeling can be fostered in the classroom. Dylan Helliwell (Seattle University) reflected on proving that the bisectors of a linear pair are perpendicular in his high school geometry class:

I couldn’t immediately put my finger on it, but this problem felt different than the others. I realized that I wasn’t solving for the measure of an angle or showing two things were congruent. I was establishing a new general fact! I was creating new mathematics! (Well, not really. Presumably the author of the textbook knew it was true, too.)

He went on to reflect more about the nature of this problem:

The statement wasn’t immediately obvious. I had to review the precise definitions and draw some examples before I believed it. Then I had to figure out the actual steps to prove it. We were using a “two-column” structure for our proofs and my proof took 31 lines! This was so much more than any of the other problems, and in the end I knew it was correct because I had proved it!

As with many research problems, this experience was significant because the student was challenged to do more than he had been asked to do before. The discovery was genuine *to him*; was new *to him*. His 31-line proof was *his proof*, and the work was meaningful because he had to think of how he could most meaningfully convey the information in those 31 lines. Tim Chartier (Davidson College) reflected on a similar experience in proving that there are infinitely many primes:

We were asked, prior to seeing the proof, to make an argument as to why we might and then why we might not have infinitely many primes. Could we run out of primes? Or, if we have some finite set of primes, is some integer large enough such that we need some new prime to form its prime factorization? Even today, I remember where I sat on campus as I pondered these thoughts. That evening, we worked on a proof of infinitely many primes in preparation for the next day’s class. In class, we developed the short proof. It was like a haiku of mathematics – elegant and focused.

This story inspires two important lessons. First, the students were not told to prove there are infinitely many primes. Instead, they were presented with the question of “are there infinitely many primes?” and asked to explore the meaning of that question. Second, the students first came up with their own proofs, and in the next class they were presented with what Erdös would call the “Book Proof” – the elegant proof that cuts to the core of mathematics. However, there was pedagogical value in this struggle against mathematics and in coming up with *a proof*, even if it was not *the proof*, because it was *their proof*. The students had ownership of the experience.

Many people who found inspiration in proving a theorem or solving a hard problem echoed a sentiment of joy in the realization that there was more to mathematics than rote calculation. José Samper (University of Miami) said

The first problem I remember enjoying was during a math competition in 8^{th}grade. I remember it well: There are 100 people on an island, some always lie, the rest always tell the truth, the islanders all know who lies and who tells the truth. A reporter comes to the island, lines everybody up, and asks the N-th person if there are at least N liars. Everybody answers “yes.” How many liars are there?

This problem made me realize that math could be more than a bunch of dull computations.

I was surprised to learn that several people had deep learning experiences as a result of rote computation. Rachel Chasier (University of Puget Sound) recalled learning her times tables:

After computing the multiples of 9 by hand, I quickly devised my own algorithm: to compute 9*N, put N-1 in the tens place and 9-(N-1) in the ones place so that the digits sum to 9. I tried explaining this to my friend, but it only made them more confused. This was one of the first times I realized I was thinking about math differently than other people and that I had a mathematical mind.

Similarly, Luke Wolcott (Edifecs Software) recalled

In early elementary school we learned about long division, and this set off a competition with me and a friend to divide the biggest numbers we could manage. I remember the passion with which I filled a 8.5 x 11 sheet, the long way, with a really big number, then drew the division bar over it and to the left, and came up with a (shorter) number to put on a piece of paper to its left. I remember the joy I felt when I realized that a list of the first nine multiples of the divisor would be very helpful, and reduced this enormous long division problem into repeated comparison and subtraction.

And finally, Lucas Van Meter (University of Washington) added

When I was in 8^{th}grade I wrote down all the squares and took their differences. I was surprised to find they were all odd. Then I took the differences of the differences and was amazed to find they were all two. Then I decided to do the same thing with cubes and finally found the differences of the differences of the differences were all equal to six. What makes this memory stick is that it was one of the first times I made a mathematical discovery on my own with no outside intervention. It felt like a personal discovery of my own.

I was surprised by these three reflections because we tend to hear that students dislike mathematics because it seems like a bunch of rote, boring computations, while these stories all seemed to stem from that rote computation. But perhaps this shouldn’t be so surprising. The important takeaway seems to be that the inspiration stemmed from discovering something new as a result of playing with all the mathematics they had at their disposal.

Finally, a number of people recalled the feeling of being struck by the simple elegance of a solution to a problem they had failed to solve. Jonathan Ke (Kamiak High School) recalled:

One day, my dad showed me a book filled with mathematical puzzles and questions, one of which was to add up the numbers from 1 to 100. I found a calculator, plugged in as many numbers as I could, got bored, and gave up. Then my dad showed me a video of how Gauss found the sum. I was amazed at the trickery he used and the mathematical explanation of why it worked. I realized that math is far more than just bunch of formulas that I choose to plug-and-chug and get an answer. It is far more complicated and beautiful.

Similarly, a colleague who works in industry wrote:

It was actually a simple problem if you knew trigonometry, but at that time I didn’t (I was in seventh grade). The obvious way to find the angles didn’t work, and I had no clue how to solve it despite a lot of effort. It turned out to be a proof by picture – just a picture – but once the teacher drew it, it was like a bright light at noon right after a pitch black midnight. The discovery was meaningful because it was the result of suddenly and deeply understanding something that you couldn’t understand before…there was joy in transitioning from being hopelessly clueless to knowing. It was one of the first times I saw that you could understand something, but first you had to make it more complicated.

In these stories, we see that failing to solve a problem can also lead to a meaningful experience. Again, the important aspect of these stories seems to be that the students had time to play with the problems first. They devoted considerable efforts to solving the problem, which led to a deeper appreciation of why the ultimate solution was so elegant.

What should these stories mean to us as teachers? On the one hand, many of them contain ideas that are prevalent in leading teaching philosophies:

- We should work to make it clear that mathematics is more than a set of arbitrary rules that govern mindless computations.
- We should create an environment in which students are encouraged to explore and share their ideas, ranging from observations about multiplication tables to new ideas about unsolved problems.
- Students need time to explore and struggle with ideas on their own before they see an elegant and perfectly rigorous solution to a problem.

So how do we do this? Some of these issues have been addressed in this blog and in other places, while others present ongoing issues to be overcome:

- How do we create a grading system in which students can be rewarded for working on a difficult problem as opposed to getting the “right answer”?
- How do we empower students to view themselves as creative problem solvers as opposed to human calculators?
- How do we assign interesting, substantive problems whose solutions cannot be found through a simple Google search?
- In many instances, people were inspired by mathematical explorations of their own design, not by problems that had been assigned to them. How can we foster this type of exploration in the classroom?

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It is commonly understood that graduate students need guidance and mentoring, especially as they begin the research phase of their studies with an advisor. A less-frequent topic of discussion is the guidance and mentoring that new doctoral advisors benefit from as they take on this unfamiliar responsibility. For many, if not most, mathematicians working in doctoral-granting departments, training and mentoring in how to be an effective advisor is done in ad hoc and informal ways. As a result, many new doctoral advisors work in some degree of isolation as they develop their advising styles.

In this article, I offer five suggestions for new doctoral advisors, suggestions that I believe make the advising process both more enjoyable and more effective. Knowledge of these suggestions can also be helpful for doctoral students, providing ideas of questions that might be helpful for them to ask their advisors.

**1. Ask About Student Career Goals Throughout the Advising Process**

It is critical for advisors to respect and support the long-term goals of their students, even when they differ significantly from what the advisor envisions as a successful career in mathematics. The range of career paths for PhD mathematicians is varied and growing, with connections to academia, industry, business, and government [1,9]. Further, the academic job market is in a state of major change and disruption [2,3]. While many graduate students begin their studies with the goal of becoming a faculty member, and while an academic career remains a reasonable goal for many current students, it is important to make sure students are aware of the reality of the mathematical job market and are informed sufficiently to make a purposeful choice about how they want to approach goal-setting.

The best way to handle these important issues is to begin the advising relationship by explicitly asking students about their career goals, discussing a range of possible goals (students do not have to pick only one possible goal!), and helping the students obtain access to any resources they need to pursue them. I have found it helpful to return to this conversation every 9-12 months, since goals often change as students gain experience and a broader vision of the mathematical community.

The critical ingredient in all of this is for advisors to ask their students a variety of questions to get a solid sense of where students are coming from, and to respond to student ideas (even if they seem undeveloped or naive) without judgement. I have personally found it surprisingly difficult to do this well — I find it reassuring to know that this is a challenge for advisors and supervisors in every field [10].

**2. Explicitly Agree on Expectations and Advising Style**

Many graduate students do not have a clear idea what to expect from the student/advisor relationship. While I have sometimes heard mathematicians complain about the ill-informed ideas or inexperience of graduate students, *this is not a deficiency on the part of students*. How are graduate students expected to acquire detailed knowledge of the graduate school experience if they are not from families that include academics or doctoral graduates? Further, how is anyone supposed to intuitively understand the expectations of a specific advisor?

It is the responsibility of each doctoral advisor to start a discussion with their students about expectations and advising style, and it is the responsibility of the student to be fully engaged in this discussion. A clarifying framework for conceptualizing advising styles can be found in the work of Gordon B. Davis in his article “Advising and Supervising” [5]. This framework describes advisor and student responsibilities for five styles: Strong master/apprentice style, Collegial master/apprentice style, Collegial development style, Guidance and suggestion style, and Passive Hands-off style. These styles are described in detail in the table provided in the Appendix below, taken from [5, Table 1]. It is reasonable to expect that at the start of the advising period, each advisor and each student will have a preferred style, possibly distinct. It is important that both students and advisors are aware of these preferences and that they are openly and clearly communicated.

It is important to observe that an advising style *does not have to be fixed* for the entire period of graduate study for a student. For example, in my own doctoral advising, I tend to start out in the Collegial Master/Apprentice or Collegial Development style (as described in the Appendix rubric) and shift over time to the Guidance and Suggestion or Passive Hands-Off style as my students mature mathematically. I do not use these phrases exactly with my students; instead, once my students have completed enough mathematical work for a complete thesis, I tell them that they should shift gears and operate more independently, pretending they are a postdoc or new faculty member. I also tell them the goal is to gain some experience in this regard before they officially start one of these roles (so far all of my students have had academic goals).

**3. Agree on a (Tentative) Plan and Review it Regularly**

Just as it is helpful for tenure-track faculty to have clear expectations regarding their tenure requirements, it is similarly helpful for graduate students to have a sense of the “trajectory” their advisor expects them to have. The best way to achieve this is to lay out a few key goals in a target timeline that is developed in collaboration with the student.

Here is an example of what I would do in one particular situation. Suppose I have a new doctoral student who has completed their written exams by the end of their second year (which is common at my institution), and who has a career goal of getting a job at a teaching-focused academic institution with reasonable but not extensive research/scholarly expectations (this career goal is also common at my institution). In such a situation, I use the following target timeline as a starting point for discussion.

Overall goal |
Graduate in May or August of fifth year — I make sure students understand that the reason for this is that at my institution, support is generally given through the end of the sixth year. Thus, in case of delays of any type, e.g. unlucky research setbacks, poor outcome on job market, etc., this provides a “backup” year that can be used if needed. |

Fall of third year |
Prepare for oral qualifying exam |

Spring of third year |
Complete oral qualifying exam, begin work on specific research problem (if not already started) |

Summer of third year |
Consider attending a summer school/program/workshop if any are available |

Fall of fourth year |
Evaluate research progress, decide if original research problem is leading to adequate progress, change problems if needed |

Spring/Summer of fourth year |
Have enough research completed to constitute a thesis the following year, write and submit one or two papers prior to beginning job search |

Summer of fourth year |
Consider attending a summer school/program/workshop if any are available |

Fall of fifth year |
Focus on job search, continue work on research |

Spring of fifth year |
Dissertation defense |

This particular timeline is not meant to be prescriptive for other new advisors, because it reflects my own advising style, the norms for my institution, the norms for my research area, etc. Rather, this timeline is meant to illustrate the level and depth of planning that I am suggesting is helpful to explicitly discuss with students. Laying out concrete goals (complete oral qualifying exam, complete first research project, etc.) gives graduate students a clear vision of what is expected from them, enabling them to better evaluate their progress through the program. Through revisiting and revising the tentative plan every 8-12 months, with input from *both* advisor and student, the planning process can be a positive and collaborative experience.

**4. Be Mindful of Mental Health Issues and the Culture of Brilliance**

As students engage in research and develop as independent scholars, it is normal for them to experience significant self-doubt and psychological setbacks. Unfortunately, these ordinary challenges are often experienced in isolation, or are amplified through a feedback loop in which their academic peers reinforce shared negative feelings instead of providing positive support. Risk of depression and mental illness is unusually high among doctoral students; however, the authors of [11] state that “research on graduate students has also shown that the quality of the advisory relationship is a significant predictor of depressive symptoms.” While mental health issues obviously involve many variables that are independent of the actions of advisors, there are concrete steps we can take to positively affect the mental health of our students.

Two of these steps are to set clear expectations and engage in collaborative long-term planning, as described in the previous two sections. As stated in [12] by the authors of [11]:

“When people have a clear vision of the future and the path that they are taking, this provides a sense of meaningfulness, progress and control, which should be a protective factor against mental health problems.”

Another step we can take is to explicitly reject the “culture of brilliance” that is frequently found in the mathematics research community [8,13]. I like the following lighthearted but on-point quote from Anne Bruton [4]:

“You do not need to be a genius to do a PhD. It certainly helps if you are bright, but some surprisingly unbright people seem to pass. The main characteristic you need in spades is ‘stickability’ — a ‘never give up’ attitude, and a willingness to suck up all problems that come your way (and they will), and find solutions to them.”

In mathematics, it is common for researchers to feel that an innate brilliance is required to be successful, even when those same researchers acknowledge that persistence and effort play a critical role. Thus, seemingly innocuous comments about “brilliant” people and “genius” ideas are ordinary and unremarkable in mathematics. In my experience, this leads to a lot of unnecessary self-doubt and loss of self-efficacy on the part of doctoral students. As indicated in [13], this culture of brilliance is also a barrier to having an inclusive and diverse research community. We must actively counter this; it is not sufficient to assume that if “we” don’t talk about this, then our students are not impacted by this facet of our culture.

A final step we can take is to show kindness and grace to our students, a topic on which Francis Su has written eloquently [7].

**5. Ask Others for Advice and Resources**

Finally, in order to help support our students, it is important that we are supported ourselves. To the greatest extent possible, new doctoral advisors should seek out trusted mentors and colleagues, whether at their own institution or elsewhere, for advice and suggestions. While some departments have effective mentoring programs, others have few formal support mechanisms in place, and junior faculty can be left in the position of needing to seek out help independently. For any faculty in that situation, reach out to others as much as possible — at every stage of our career, we each benefit from mentoring and support.

**Acknowledgements**

Thanks to my own doctoral advisor, John Shareshian, for always caring about the well-being of his students personally as well as mathematically. Thanks to Carl Lee, my mentor when I was an Assistant Professor, for his guidance when I was a new doctoral advisor. Finally, thanks to my current and previous doctoral students, all of whom have made my mathematical work more meaningful.

**Appendix: Rubric of Advising Styles**

Style |
Advisor Role and Behavior |
Student Role and Behavior |

Strong master/apprentice style | Advisor is master. Advisor has a well specified domain of expertise and set of problems within it. | Student is an apprentice working for the advisor. Student works on advisor’s problems. |

Collegial master/apprentice style | Advisor is expert who limits advising to problems that are within scope of his or her research skill set but will work on student’s problem. | Student develops a problem within advisor’s domain and skills and works under the advisor to develop the research plan and procedures. |

Collegial development style | Advisor is senior colleague who will respond to student research problem and extend his or her advising domain to include new problems and new skills. | Student takes initiative to introduce new problem that requires new skill set and works as a junior colleague with advisor in joint development of new domain. |

Guidance and suggestion style | Advisor is a senior colleague who gives good general guidance over a wide range of problems and methods but does not have personal skill in all of them. | Student is an independent, junior colleague who takes initiative for presenting problems and research plans for discussion and guidance. Student develops required skills. |

Passive hands-off style | Advisor has quality control role and responds only to requests or documents and performs only general quality control review. | Student is an independent researcher who takes initiative for developing problem, developing skills, and presenting research plans for general review and approval. |

**References**

[1] “Math PhD Careers: New Opportunities Emerging Amidst Crisis.” Yuliy Baryshnikov, Lee DeVille, and Richard Laugesen.* Notices of the American Mathematical Society*, Vol 64, No 3, March 2017, pp 260-264 http://www.ams.org/publications/journals/notices/201703/rnoti-p260.pdf

[2] “Survey on Math PostDocs.” Amy Cohen, Letter to the Editor, *Notices of the American Mathematical Society*, Vol 64, No 6, June/July 2017, pp 541 http://www.ams.org/publications/journals/notices/201706/rnoti-p540.pdf

[3] “Disruptions of the Academic Math Employment Market.” Amy Cohen. *Notices of the American Mathematical Society*, Vol 63, No 9, October 2016, pp 1057-1060 http://www.ams.org/journals/notices/201609/rnoti-p1057.pdf

[4] “Dear new PhD student — a letter from your supervisor.” Anne Bruton. Weblog, https://anniebruton.wordpress.com/2013/09/21/dear-new-phd-student/ Retrieved 19 December 2017.

[5] “Advising and Supervising.” Gordon B. Davis. In *Research in Information Systems: A handbook for research supervisors and their students.* Butterworth-Heinemann, 2005. Preprint at http://misrc.umn.edu/workingpapers/fullpapers/2004/0412_052404.pdf

[6] “Advice to a Young Mathematician.” Sir Michael Atiyah, Bela Bollobas, Alain Connes, Dusa McDuff, and Peter Sarnak. In *The Princeton Companion to Mathematics*, Ed. Timothy Gowers. Princeton University Press, 2008. http://assets.press.princeton.edu/chapters//gowers/gowers_VIII_6.pdf

[7] “The Lesson of Grace in Teaching.” Francis Su. Weblog, http://mathyawp.blogspot.com/2013/01/the-lesson-of-grace-in-teaching.html Retrieved 22 December 2017.

[8] “Belief that some fields require ‘brilliance’ may keep women out.” Rachel Bernstein. *www.sciencemag.org*, 15 Jan 2015. http://www.sciencemag.org/news/2015/01/belief-some-fields-require-brilliance-may-keep-women-out Retrieved 22 December 2017.

[9] National Research Council. *The Mathematical Sciences in 2025*. Washington, DC: The National Academies Press, 2013. http://www.nap.edu/catalog/15269/the-mathematical-sciences-in-2025

[10] *Humble Inquiry: The Gentle Art of Asking Instead of Telling*. Edgar H. Schein. Berrett-Koehler Publishers. 2013.

[11] “Work organization and mental health problems in PhD students.” Katia Levecque, Frederik Anseel, Alain De Beuckelaer, Johan Van der Heyden, Lydia Gisle. In *Research Policy,* Volume 46, Issue 4, 2017, Pages 868-879, ISSN 0048-7333. http://www.sciencedirect.com/science/article/pii/S0048733317300422

[12] “Ph.D. students face significant mental health challenges.” Elisabeth Pain. *www.sciencemag.org*, 4 April 2017. http://www.sciencemag.org/careers/2017/04/phd-students-face-significant-mental-health-challenges Retrieved 24 December 2017.

[13] “Expectations of brilliance underlie gender distributions across academic disciplines.” Sarah-Jane Leslie, Andrei Cimpian, Meredith Meyer, and Edward Freeland. *Science*, Vol. 347, Issue 6219, pp. 262-265. http://science.sciencemag.org/content/347/6219/262

I’ve often thought that we could do a lot better job of explaining “advanced” mathematics concepts in simple language for the benefit of a wider audience. As a student, I never liked being told, “We’ll explain that to you next year.” As a teacher, I’ve always wanted be able to give real answers to students’ exploratory questions: if a Calculus I student asks me a question whose precise answer requires knowledge of manifolds and de Rham cohomology, I want to be able to distill those ideas into an answer that this student can understand. Also, I’ve always enjoyed the challenge of telling non-mathematicians about Euler’s formula, or voting theory, the Four-Color Theorem, or the game of Nim. I have experimented with trying to explain my own research in algebraic combinatorics to an intelligent layperson.

For example, I recently coauthored a paper with the intimidating title “Oscillation estimates of eigenfunctions via the combinatorics of noncrossing partitions”, which is full of extensions of eigenfunctions, fractional Schrödinger operators, Kreweras complementation, and similar jargon. I summarized it like this: “My coauthors study partial differential equations, which model things like fluid flow and heat dispersion. They draw pictures that look like tangled-up spaghetti, then try to measure the complexity of the equations by counting the holes in the tangle. Well, counting is what I do for a living, and when I saw their pictures, I was able to use what I know about counting to help them solve their problem.” Sure, that’s sweeping a whole lot of things under the rug, but really, that’s what we were doing.

And that is how I ended up editing articles about mathematics for kids.

Frontiers for Young Minds is an online, open-access scientific journal. Authors write articles about topics they are experts on and submit them to the journal; the editor assigns a reviewer to read the article and submit comments, and works with the author to address them, and ultimately makes a decision about publication. Sounds familiar, right? There is one big difference. At this journal, the articles are *not* peer-reviewed. They are *kid*-reviewed. As you may have guessed from the title of the journal, the target audience consists of kids (ages 8-15), and who better than a prospective reader to tell the author what is working and what needs to be fixed?

I heard about Frontiers about a year ago from a biologist friend. The journal was founded in 2013 and has several established sections: Astronomy and Space Science, Biodiversity, Earth and its Resources, Health, and Neuroscience. Where was the Mathematics section? I wrote to the editor-in-chief (Robert Knight, a neuroscientist at UC Berkeley): “Why isn’t there a section on mathematics?” I got the response I expected: “Why don’t you start one?” Since then, we’ve put together a board of editors and Science Mentors (who assist the reviewers) from all area of mathematics for this newest section of the journal, which we’re calling Understanding Mathematics. We are about to publish our first article, on game theory, and we are encouraging submissions. I hope that you will consider contributing.

**How can I get involved?**

Write an article! Join the editorial board! Be a Science Mentor! Please! But especially, write an article — we need content!

**What makes a Frontiers article?**

They fall into two categories: “Core Concepts” (think Euler’s polyhedral formula, or the infinitude of the primes) and “New Discoveries” (your latest research breakthrough). Articles are short (2000 words, approximately 4-5 pages) and should be fairly self-contained. It is okay (even encouraged) to suppress all the technical details; remember who the audience is.

**Why should you write an article for Frontiers?**

Because mathematics is fun and interesting and beautiful. If you’re reading this blog post, you know that. But you also know that non-mathematicians frequently perceive mathematics as tedious and boring and hard. And a lot of that attitude probably starts around ages 8-15 or so. So let’s try and change that. Why don’t we try explaining something wonderful like Euler’s polyhedral formula, or why there are exactly as many even numbers as counting numbers, or why there are infinitely many primes, or what the fourth dimension looks like? Why not communicate to kids that mathematics is a living subject, that we are constantly making new discoveries — think Fermat’s Last Theorem, the Poincaré Conjecture, the twin primes problem, or your own most recent paper?

**How do I write for this audience?**

Most of us mathematicians and scientists are used to an audience of fellow experts, even specialists. We write for readers who share our background knowledge and speak our lingo. We assume that they are comfortable with definitions, proof techniques, fancy notation, and all the other tricks of the trade. Writing an article that will be intelligible to 8- to 15-year-olds — especially the young end of this range — requires different strategies. The Frontiers editorial staff has a whole lot of suggestions about how to write for this audience. Here are some highlights:

- “The goal is not to explain as much of your work as can possibly fit within ~2000 words, but rather to craft a clear and self-contained explanation that covers all that a novice reader for your target age group might need to be able to understand your work.”
- “If something is exciting, convey that excitement. If something is important, say exactly why it is important rather than burying that value within a paragraph of inferences and implications.”
- “Unlike an academic abstract, which is meant to serve as a dispassionate summary of the academic content within an article, these abstracts are meant to draw readers in.”

Indeed, aren’t these principles are just as valid in communicating to our colleagues and our students as they are in writing for young readers?

**What’s a Science Mentor?**

Science Mentors work directly with the young reviewers to explain what peer review is, guide the reviewers in reading and critiquing the submission, fill them in on any mathematics or science background they need, and act as the reviewers’ advocates in the next stages of the editorial process. If you’re a graduate student or a postdoc, this is a great way to get involved with Frontiers for Young Minds.

**What’s the review process like?**

Well, it’s similar to a “standard” journal: author submits article, reviewers give feedback, author revises article, and the process repeats until everyone is happy with the final version (as the shampoo bottle says, “lather, rinse, repeat.”). One difference is the interactive review forum, in which the author, the handling editor, and the Science Mentor (acting as the reviewers’ advocate) can communicate directly about revisions. Some articles are reviewed by one or two kids, some by an entire classroom. Sometimes the kids can be disarmingly frank (here’s one quote I like, from some years ago: “This seems important, but the way it is written is so boring I can’t even get to the end. Could the authors maybe sound excited about what they are doing?”)

David McAdams, the author of our first article, had this to say about the review process:

“The kids’ feedback is surprisingly helpful. Part of this is due to the Science Mentor, who elicits the most useful comments and then filters what the kids have to say, but I think it’s also due to the fact that the kids work as a team in reading and evaluating the paper. It’s as if an entire research group studied and debated your paper before preparing a jointly-authored referee report — much more useful and effective than the usual model of sole-authored reports.”

Even with authors and editors who are trying in good faith to write an article that is self-contained and readable by 8- to 15-year-olds (or a subrange of ages), it is easy to make incorrect assumptions about readers’ background knowledge. Sometimes these assumptions can be quite subtle. We need the reviewers to pinpoint these issues for us. It’s one thing to ask yourself how a 10-year-old reads; it’s another thing to *be* a 10-year-old. David McAdams reported that the kids, through the Science Mentor (graduate student Amanda Wilkens), had given him some crucial feedback: there wasn’t *enough* mathematics in the article. “I had gone a bit overboard in terms of being non-technical, to the point that mathematical arguments didn’t actually seem like math! I corrected this by adding a specific example with numbers, and the paper is much improved as a result.”

I not only hope that the journal will be a resource for kids who want to learn about mathematics, but also that writing a Frontiers for Young Minds article will be a rewarding experience for authors: here is an opportunity to communicate your work more broadly, not just for fellow specialists. Scientists who have submitted articles to the journal often tell the editors how anxious they were to receive the kids’ feedback: it can be a challenging task but a very satisfying one. (One author told us that he was pleased to have written an article that people might actually read!) I hope you’ll consider helping us grow by submitting an article on your favorite topic of mathematics.

]]>Mathematics is a beautiful subject that can easily become an ivory tower. There can be a temptation for teachers and students of mathematics to shy away from the role that mathematics plays as a social force and a barrier that can put a halt to a person’s career, security, and social mobility. The mathematics education community has been studying this situation for years – for example, see this article by Rochelle Gutierrez [1]. One way to include a focus on society and its problems in a mathematics classroom is by introducing service-learning into one’s course.

Service-learning is a pedagogy that combines the course content with community service that is directly tied to the material that students are studying inside the classroom. Service-learning has traditionally belonged to the domain of social sciences such as psychology, sociology, or social work, however interest in service-learning has recently increased in STEM disciplines as well. A special issue of PRIMUS [2] was entirely dedicated to mathematical service-learning projects; an interested reader will find a wealth of helpful practical information and project descriptions there, from math fairs and tutoring to running modeling projects for community organizations. In this post, I would like to share with you my own experience with service-learning, its effect on my students’ worldview and mathematical knowledge, as well as offer some suggestions for the instructor who would wish to introduce service-learning into a math course.

When I first learned about service-learning four years ago, I immediately wanted to try it – and my initial motivation was practical. Precalculus students are a mathematical population at risk. Weak algebra preparation invariably hinders progress of STEM students, and severely affects performance in Calculus, a major junction in the leaky STEM pipeline. As teachers, we know that one of the best ways to learn something is to teach it ourselves: “I hear and I forget. I see and I remember. I do and I understand”. This led me to ask myself: “What if university students in my classroom had to teach algebra prerequisites to someone else? Will it help them learn and understand that material themselves?”

Since 2015, I have implemented service-learning as a component of one of my winter precalculus classes, each section numbering 21-25 students. In addition to the regular lecture-homework-tests-problem-solving part of the course, students spent 2-3 hours each week tutoring mathematics to students in the community, an urban environment with a large immigrant and refugee population. In the past three years, students have served in local middle schools, a high school, a community college, and an elementary school. My goal was to place students at or above the middle school level, so that the mathematical content would be sufficiently challenging; the option to tutor at the elementary school was reserved for exceptional situations.

At the end of the course, I saw interesting results in my students’ course performance. During the first iteration in 2015, I saw a reduction in fundamental mistakes made on the final exams between service-learning and regular sections. Another time, a less-prepared service-learning group caught up in exam performance with the initially stronger regular section, and demonstrated a positive shift in beliefs about importance of conceptual understanding in mathematical problem-solving. The full description of the first iteration of the service-learning experiment and its results appeared in the RUME XIX Conference Proceedings [3].

In general, the atmosphere in the service-learning course seems different, with students being more focused and rarely slacking in their own work – perhaps because they feel they are role models for their own students. This is quite consistent with past experience of my colleagues who also found that service-learning produced an interesting effect in their classroom. For example, Allison Henrich [4] noted a decrease in math anxiety in service-learning students taking a math course for non-majors.

Reflection is an integral component of a service-learning course, because it serves as a tool for converting experience into knowledge (thank you, Jeffrey Anderson, for a great analogy!). In my class, students engaged in structured reflection to help them process their experiences. Twice during the quarter, we spent 40 minutes of class time talking about the students’ service-learning experiences. By talking to classmates who worked at the same location, students built a community and a support network, and resolved practical problems. The class discussions focused on questions raised by students themselves: “How would you explain to a student what a variable is?” “What do you do if someone is disrespectful?” “How do you motivate an unmotivated student?”

Students also kept a weekly reflection journal, and reading it was one of the most rewarding – and time-consuming – parts of my experience as instructor of a service-learning course. Each entry consisted of a mathematical and non-mathematical reflection. In the mathematical part, students analyzed their teaching interactions and reflected on the subject material that came up in the tutoring sessions. Students often noted parallels between the material we were covering in our class and their tutoring sessions, where they explained concepts of graphing and solving equations, and organizing data for word problems.

The optional non-mathematical part was free-form, and students could reflect on the human part of their experience, describing the kids they observed or the difficulties that immigrant students faced. One student reflected:

Oftentimes when a student is confused in class, it’s not that they don’t understand math, it’s that they are struggling translating the English being spoken.[…] If I wrote the problem out using numbers they could solve it immediately. They are astonishingly intelligent.

Quite often students saw themselves in the students they taught, which led them to recognize their own mal-adaptive mathematical strategies, such as not reading a problem or rushing through a solution. Some drew strength and inspiration from the personal struggles of their students, regardless of age, and witnessed how difficult life circumstances combined with poor math preparation put educational and career goals out of reach for their students.

Tutoring at Seattle Central Community College is really eye opening. It’s really made me think of my blessings and truly be grateful for them…Recently one of the people I tutor (let’s call her Hana) has dropped out of school. This was a woman that I admired and looked up to from the moment I met her. […] When she told me of her decision over the phone, I was surprised by how big the blow was to me. She was a 57 year old woman aiming to be a nurse. She showed me how determined a person can be even when the odds and/or circumstances are against them. She had dropped out due to medical reasons which really upset me due to how well she was picking up on her material in class.

As the instructor of a service-learning course, one should be prepared for a time-consuming, rewarding, often intense, sometimes draining, unexpected and interesting experience. One should be aware that service-learning has potential to lead to lower overall course evaluations, despite the overwhelmingly positive student feedback on the tutoring experience. Service-learning introduces a potential for instability into the classroom, and the instructor has little control over the way this experience will unfold. I found myself addressing all sorts of matters, from practical to political. Seattle University has a wonderful Center for Community Engagement, which is instrumental in organizing student placement, arranging background checks, finding community partners, as well as providing cultural competence training to students. If a university does not have such an organization within its structure, the necessity to find community partners and build connections falls to the instructor, which is a large and time-consuming endeavor.

The instructor in a service-learning course has a responsibility to carefully frame the discussion about the experience so that service-learning does not end up reinforcing existing stereotypes and instill a “savior complex” in participating university students. The instructor should also provide some pedagogical training to students and discuss helpful tutoring strategies and the necessity to teach for understanding. Some pedagogical training can happen during regular lectures, while other discussions will fit well into the in-class reflection sessions.

Hopefully, the mathematical benefits can convince the more traditional and conservative math departments to implement some form of service-learning in their classes. A liberal arts or religiously-affiliated institution may follow a mission statement that aligns with a message of service and social justice, and so may be more open to service-learning as pedagogy. I would like to end this post with a quote from student reflection:

I believe that service-learning should be something that everyone should at least participate in at least once. Maybe it will be a hit or miss, but if it is a hit with some individuals, they can definitely devote their passion and their drive to teaching scholars and students who would greatly benefit from tutors.

[1] Rochelle Gutiérrez. (2013). The Sociopolitical Turn in Mathematics Education. *Journal for Research in Mathematics Education,* *44*(1), 37-68.

[2] Special Issue on Service Learning. *PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies* 23:6

[3] Ekaterina Yurasovskaya. (2016) Service-learning in a precalculus class: Tutoring improves the course performance of the tutor. In (Eds.) T. Fukawa-Connelly, N. Infante, M. Wawro, and S. Brown, *Proceedings of the 19th Annual Conference on Research in Undergraduate Mathematics Education*, Pittsburgh, Pennsylvania. 496–505.

[4] Allison Henrich, and Kristi Lee.(2011) Reducing math anxiety: Findings from incorporating service learning into a quantitative reasoning course at Seattle University. *Numeracy.* 4(2).

In one of life’s weird coincidences, when I moved to a small town in East Texas to start my academic career at Stephen F. Austin State University (SFA) ten years ago, I didn’t know that I would be working with someone from my high school graduating class of about 150 people. Through that small quirk of life, I met a lot of the art faculty and local artists in the Nacogdoches area. I love that I get to hang out with artists and art educators. They are really cool people and they *MAKE THINGS*. Things that people want to look at, things that people want to discuss, and sometimes, things that people even buy.

The idea I want us all to consider is: *“How do we grow and improve the culture of making and improving things for our teaching?”* When I say things for our teaching, I mean much more than just textbooks and notes for lecture; I mean software and technology that adds meaning and value for our students; I mean activities that can change the attitudes, habits, and practices of our students; I mean the many other materials and resources that will transform students and mathematics classrooms. While I will cite some examples below of projects and resources that I think are doing good things, I genuinely think we need to **not** compartmentalize these practices, but make them part of what *we* as a community do.

**The Teacher as Artist**

Anyone who knows me and has seen me since last May knows all about my most recent project because I can’t contain my excitement for it. My SFA colleague Jeremy Becnel and I spent the summer of 2017 building an app that uses a Google Cardboard viewer and a smart phone to allow the user to visualize the concepts of multi-variable calculus in a virtual reality (VR) setting. You can see more about what we have accomplished and where we are going with this project here. It has been a frustrating, wonderful, humbling, and exhilarating process. Jeremy has taught me a lot about writing good code for software so that we have less of this kind of thing:

I could spend the next ten thousand words just talking about the problems and solutions, both great and terrible, we encountered so far on this project, but I am saving that for a grant proposal.

By far, the thing I have liked most about this project is the opportunity to make something that I think is beautiful and interesting and useful and new. While I like thinking about traditional mathematical research questions, I struggle with that type of work. Not just because mathematical research is hard, but I sometimes feel like I’m doing it because someone told me I need to do this type of work to advance to the next level. There is tremendous value gained by pushing through the struggles of understanding and expanding the frontiers of mathematical knowledge, but I bet I talk to more people in a month about my virtual reality project than I ever have about my research in dynamical systems. Those conversations about virtual reality and the process of making things has been a breath of fresh air for my career and my enthusiasm for my work. It certainly helps that this project lets me draw pictures and see patterns (i.e., *do mathematics*) in an immersive environment that seems to be of broad interest to a mainstream audience.

I love having something that I can show other people that leads to a discussion like the ones I enjoy having with artists. I usually get asked something along the lines of “How did you make that?” or “How are your students using this?” When I meet an artist for the first time, I usually ask similar questions like “What does your creative process look like?”, “How do you decide when a piece of work is done?”, or “When you look at art, how do you evaluate it?” I have discovered that artists make pieces for lots of different reasons and with different motivations. Having created a tool for teaching, and being on the receiving end of these types of questions, makes me feel like more of an artist than my traditional research has. In other words, the value of making things is not only the creation of the product but also the discussion that is prompted by the process of making and evaluating the product.

**Sharing the Art in Teaching**

When I reflect on the VR project, I think that there were a few things that were vital to the early successes we have had. First, we had a clear vision of what we wanted the product to look like and what we wanted the materials to do. It is so easy to get caught up in the minutia of making something that you forget about its purpose and audience. Second, there were some tools available that helped bring down both the cognitive and technological hurdles. (Thanks Unity, Unity Development Community, and Google VR!) Third, it built upon the habit I had long been practicing of making things of my own for teaching. Those things were often terrible at first. Some of them came from half-baked ideas and others came out of a curiosity for whether I could even do the thing. Some of these terrible things I made showed me how to make things better, more useful, or more aesthetically pleasing. Depending on how urgently busy I felt at the time, I would modify the stuff that I was making. Lastly, we had time to wander. Let me be more specific. One of *The Five Elements of Effective Thinking* by Burger and Starbird is to follow the flow of ideas, for better or worse. This was our unstated motto for the first month of development in the VR project. I found how wonderful it was just to explore what you can create and how polished you can make it.

Over the past few years I have become more active in the Inquiry Based Learning (IBL) community and in my work with Project NExT in the Texas Section and at the national level. Both endeavors have been very useful for me as an individual, but more importantly, they are both building communities and improving the cultural practices of our profession. Stan Yoshinobu gave a great talk at MathFest on “The Next 20 Years: Moonshot Challenges in Post-Secondary Math Education and IBL.” Stan also outlined goals and steps to reach those goals, as well as ways to build groups to address these steps. One of the most appealing aspects of the “big tent IBL” that Stan talks about is the idea of transforming students from users of information into creators of information. A great strength of the Project NExT groups is the ability to expose new faculty to ideas and resources that can help them evaluate what their role is in the classroom, in their departments, in our profession, and in an increasing complex and demanding world. Further, Project NExT offers a community of continuing support for faculty to strive to become the best versions of themselves.

Currently, there is a lot of work, both in academia and in the larger world, on identifying and dismantling barriers to progress in education (and society in general) that are arbitrary, hidden, or intentionally obstructive. For instance, the cost of textbooks and software is certainly an obstacle, but it’s one our mathematical community has been working to remove. We have many tools now that make publishing and sharing resources incredibly easy. As a community, we have made great tutorials on using open source, freely available mathematical software like Sage. Projects like the AIM Open Textbook Initiative have started to centralize and vet valuable resources. Projects like UTMOST are aimed at growing both the community and the tools for open source textbooks. We need to continue to examine the hidden and arbitrary barriers and costs of building great resources for teaching, as these projects do.

Just as we try to model good behaviors and practices for our students, we should make sure we are modeling good behaviors and practices for our new faculty. Do we value and support making quality materials for teaching in a comparable way to the way we value and support quality research? Further, I think we must minimize the “tyranny of the urgent” on new faculty to allow them to think bigger about their goals for teaching and making things for their teaching. Urgency is the antithesis of strategy, and we must properly harness the energy, ideas, and enthusiasm of our new faculty for teaching. Good management and administration allow all of us to flourish in all aspects of our jobs.

For example, many departments are able to offer teaching load reductions during the first year to allow new hires to adjust to new expectations. I have benefitted from this kind of accommodation for the purposes of research as well. Unfortunately, the support resources for building curricular materials can be spotty at a local level. Larger scale projects involving the making of teaching things, like UTMOST and WebWork, have received support from places like NSF and MAA. These big projects had to start as small projects and without some local support, they could not have grown to what they are now. We should be pushing for more resources at both the local and national levels for support to build things related to teaching. It is also imperative to increase the collaborations between mathematics and mathematics education faculty to build, measure, and improve upon the things we make for our teaching. Building these connections should be an explicit part of the many existing faculty communities like Project NExT and should be considered through many regional organizations like MAA and AMS Sections.

**Conclusion**

Francis Su’s 2017 MAA Retiring Presidential Address “Mathematics for Human Flourishing” does a wonderful job of talking about doing math as a truly human endeavor. “A Mathematician’s Lament” by Paul Lockhart is an insightful essay contrasting the way teaching comes through to students across disciplines. Lockhart’s work even cites G.H. Hardy’s description of mathematics as art:

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with

ideas.

Both Lockhart’s and Su’s pieces examine aspects of the culture of teaching mathematics and the culture of who should do (make) mathematics, respectively. I have been thinking a lot about how I make things for my teaching, how much of this making is an individual practice and how much of it comes from the culture of teaching in mathematics. There is a truism from business management “Culture eats strategy for breakfast.” Culture is a representation of what “we” do. Strategic planning and vision statements come and go, sometimes at mind boggling speed, but people (mathematicians are people too…) are loyal to culture and not to strategies. Culture is about the expectations and corresponding accountability for all in a community. And I hope that we can find a way to grow our culture of making remarkable things for our students and their learning environment as pervasive and encompassing as the culture making of art is for artists.

]]>The more I teach and learn mathematics, the more I regard the subject as a powerful resource that is unfairly distributed. Clearly, I’m not alone. Search for “underrepresented” on the American Mathematical Society website and you’ll find the inclusion/exclusion blog and the Director of Education and Diversity at the AMS, for example. While it is vital to build on the work of exemplary programs at the university level, we cannot fully address the inequities in access to mathematics, and to fields that require mathematics, unless we also examine and address inequities in pre-college education.

Like many colleges and universities in the U.S., Middlebury College, where I teach, has made clear progress toward serving a more diverse student body over the past two or three decades. We’ve come to understand that it’s not enough to admit students with varied backgrounds and experiences; the institution has a duty to support all of its students once they arrive. The higher education community is still figuring out what forms of support are most promising, and some interesting models have emerged. For example, Middlebury is one of many partner institutions with the Posse Foundation, which selects students who “might have been overlooked by traditional college selection processes,” and provides training, mentoring, and peer support networks. The Meyerhoff Scholars Program at the University of Maryland, Baltimore County works with undergraduates interested in pursuing doctorates in science and engineering, and “persistence models” are aimed at reducing high attrition rates of female and underrepresented minority (URM) STEM majors.

Within the college mathematics realm, the Emerging Scholars Program (ESP) was created in the 1970’s at the University of California at Berkeley, and has been adapted at many other institutions [3]. Another program well known for welcoming many students, including women and URM students, is the mathematics department of the State University of New York at Potsdam [1].

Judging from the instantaneous standing ovation after Francis Su spoke of “Mathematics for Human Flourishing” at the Joint Mathematics Meetings in January, I know that at least hundreds of mathematicians are deeply concerned that, as Su said then, “(t)he demographics of the mathematical community does not look like the demographics of America. We have left whole segments out of the benefits of the flourishing available in our profession.” Some of us have tried to address the issue in our own departments by emulating ESP, SUNY Potsdam, and other established programs. Some of us have taken a close look at our own classrooms and found promise in active learning, given growing evidence of its effectiveness for all students, not just the over-represented, and are heartened by continuing research and development of active learning environments. Some of us have worked toward justice in undergraduate mathematics in other roles as mentors, administrators, and writers. This work must continue. We need to broaden and refine our understanding of inclusion on our campuses.

At the same time, it is impossible to ignore the inequities in our students’ K-12 experiences. The evidence is right in front of us, in the form of large variation in the strength of our students’ mathematical backgrounds. Recently a Middlebury sophomore told me of her AP Calculus class in an urban public school: everyone in the class scored a 1 or a 2 on the exam, which is typical for that school. This simply would not happen, for example, at the public high school I attended, which now offers AB Calculus, BC Calculus, and Multivariable Calculus. Because public education in this country is, in large part, funded locally, students in high-poverty districts face constraints that their peers in wealthy suburbs do not. Some rural school districts are unable to keep qualified teachers, for example. Evidence suggests that teacher turnover is higher in less affluent communities, with more negative results. I have come to realize that we can’t expect to close, in four years, gaps that have been growing for twelve years or more. Later in this post, I explore some of the ways mathematicians are engaged in addressing those gaps at the pre-college level, and welcome additions to that list.

First, let’s examine the problem a bit more closely. For example, we know that mathematics is an important part of preschool education. We also know that access to high-quality preschool programs is unevenly distributed; in fact, a startling percentage of 3- to 5-year-olds in the U.S. don’t attend preschool at all, and that those children are disproportionately from low-income families. Thus the educational disparities correlated with economic inequality start before kindergarten.

The challenge is not simply one of poverty, though. It’s impossible to talk about inequity in our pre-college educational system without considering race. A “Dear Colleague” letter from an officer at the U.S. Department of Education in 2014 offers ample evidence that

(c)hronic and widespread racial disparities in access to rigorous courses, academic programs, and extracurricular activities; stable workforces of effective teachers, leaders, and support staff; safe and appropriate school buildings and facilities; and modern technology and high-quality instructional materials … hinder the education of students of color today.

Meanwhile racial segregation in schools persists. Nikole Hannah-Jones has documented this phenomenon in extensive work that recently earned her a MacArthur Genius Grant. This fact, together with continuing confirmation that separate is not equal [2], is a reminder that inequities are deeply entrenched. I struggle with these disheartening realities, but end up in favor of doing something rather than doing nothing. We in the mathematics community have specific expertise and resources to offer. The challenge, of course, is figuring out how to put them to good use.

Professors of mathematics searching for ways to take on educational disparities in schools have several models to follow. The National Association of Math Circles has been working to provide more support for Math Circles that serve URM students. The Navajo Math Circle project is one inspiring example (if you haven’t seen the film, do!), and Math CEO at the University of California, Irvine is another. For an example of a program designed to intervene early and support students over time, consider Bridge to Enter Advanced Mathematics (BEAM), whose mission is to “create pathways for underserved students to become scientists, mathematicians, engineers, and computer scientists.”

Another option is working with practicing teachers. There are Math Teachers’ Circles,for example. My interest in K-12 education led me to working with practicing teachers in a master’s degree program, and many of our colleagues do similar work. The Benjamin Banneker Association, an affiliate with the National Council of Teachers of Mathematics, supports teachers of African-American students.

Finally, mathematicians at various colleges and universities teach mathematics to future teachers. We should recognize the importance of this work, and reconsider the diminished status it has in some quarters. We should also honor those who weave equity considerations into their classes for future mathematics educators.

Please let me know what I’ve missed! My co-editors Ben Braun and Diana White suggested some of the examples above; I know there are more. Mathematicians in higher education and industry are not always experts in preschool, elementary, or secondary education, but the current situation calls us to supporting roles in the pursuit of equity. Have you, or people you know, taken on such roles in promising new ways? Comment below, or send me an email. I look forward to hearing from you.

REFERENCES

[1] Hersh, Reuben. “The Best Undergraduate Math Program You’ve Never Heard Of.” *Math Horizons* 17, no. 3 (2010): 18-21.

[2] Mickelson, Roslyn Arlin, Martha Cecilia Bottia, and Richard Lambert. “Effects of school racial composition on K–12 mathematics outcomes: A metaregression analysis.” *Review of Educational Research* 83, no. 1 (2013): 121-158.

[3] Murphy, Teri J., and Uri Treisman. “Supporting high achievement in introductory mathematics courses: What we have learned from 30 years of the Emerging Scholars Program.” *Making the connection: Research and teaching in undergraduate mathematics education* 18, no. 73 (2008): 205.

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*Editor’s note: The editors thank Stacie and Matthew for taking the time to share their thoughts and insights with us about their REU experiences. For students who are interested in applying for an REU, lists of programs can be found **here** and **here**. To read more about student and faculty experiences with REUs, see our other articles on this topic.*

**Stacie:**

I did not know what a Research Experience for Undergraduates (REU) was until one of my professors suggested I apply to REUs at a few different institutions. I went to a small liberal arts college with few research opportunities. I applied to a handful of different REUs and was excited when I received my acceptance from Marshall University. I was excited to spend eight weeks of my summer at a different institution and to learn what mathematical research really looked like. I was also nervous that I would be behind the other students academically. When I arrived, my nerves were calmed and the excitement continued.

**Matthew:**

If I learned anything from participating in the REU at Marshall University, it was how to be frustrated. Before the REU, I had certainly encountered a few difficult proofs in my courses, some of which I spent a couple of days thinking about. However, I had never spent an entire summer obsessed on one seemingly tiny mathematical problem. Throughout the months of June and July, I thought about mathematics at breakfast while drinking my coffee. I thought about mathematics while sitting in a classroom at Marshall. I thought about mathematics while playing cards in the evening. I went as far as to buy a 3-foot-wide whiteboard to keep beside my bed that I could grab in the middle of the night and test out propositions.

When I describe that experience to my friends outside of the mathematics community, they usually say, “That sounds awful”. But in actuality, it was the most exciting experience of my brief mathematical career. In a short span of time, I learned a great deal of exciting new mathematics that I would not have been exposed to in a normal course at my home institution. I learned a great deal more about the mathematical community in general, how research works and where I might fit in. Most importantly, I learned how exhilarating the process of figuring things out mathematically can be.

**Stacie:**

I have had a few small research-based projects in different classes. However, none of them compare to my experience at the Marshall University REU. I got to spend eight weeks with nothing to worry about except math. Unlike the other projects, I did not have any other schoolwork or activities to complete. All I had to worry about was the research. I found myself thinking about it at all times of the day. I would lay down to go to sleep and have to get up to write down an idea that came to me. I would be at the gym working out and I would think of an idea. I found it funny how I could spend all day working on something and then whenever I started to do something else the solution would pop into my head. I enjoyed working on something no one had ever worked on before. Unlike homework, there was no obvious correct solution and my adviser did not always have all the answers. My adviser was able to lead us in the correct direction most of the time, but because he did not have the answers sometimes he was as stuck on a problem as we were. I felt a great sense of accomplishment when I was able to prove something new. At the end of the summer we proved the result we set out to prove and I was extremely proud of the rest of my group and myself.

**Matthew:**

In the middle of the summer, the work I was doing with my group had nearly slowed to a halt. We had a good deal of success in June, but by July, we really did not know how to proceed. I was spending almost every free moment that I had in the Marshall Graduate Student’s office trying to parse through sheets and sheets of output from Mathematica programs, looking for patterns. It seemed like every morning that our research group met, I would have a new proposition to test, only to find a counterexample before going to lunch. It was immensely frustrating, especially because some of the other research groups within the REU seemed to be making so much more progress than us.

However, all of the frustration was worth it when I finally made a minor breakthrough and found a combinatorial pattern that I thought might be the basis of a future result. When I found what I had been looking for, it seemed like it had been blatantly obvious, yet it had taken me weeks of work to find it. The moment was exhilarating. Whole new avenues of inquiry were suddenly opened up, and for a week, the project looked extremely promising.

Then I got stuck again. But that’s the beauty of it. Because of the experience I had in the REU, I knew that with enough hard work and creative thinking, I would eventually make another small step in the right direction. It was like hiking in unfamiliar territory. Most of the time, I was walking in an almost random direction, completely unsure of whether I was on the right path or not. But then, I would see a sign and know which way to turn.

The most important thing the REU provided was the opportunity to take risks. Because there are no grades, I had the freedom to focus very intently on a single topic. For once, it was okay to fail, okay to suggest a proposition that I would find a counterexample for a few hours later. In most of the mathematics courses I have taken, the focus has been on getting the correct answer in the most efficient manner possible. But the most efficient manner excludes all of the wrong turns and stumbles that someone had to make to get to that point. Through my experience at the REU, I was able to explore the un-polished bits of mathematics that don’t make their way into our textbooks.

**Stacie:**

My experience through the REU has strengthened my ties to the mathematical community. Throughout the summer I met many professors from different universities and I got to meet fellow students who have similar career goals as me. The REU gave me the opportunity to travel to the Joint Mathematics Meetings in Atlanta, Georgia. It was amazing to see so many mathematicians in one place. I got to listen to experts talk about fields of study I am interested in and I got to meet many different undergraduates, graduates, and professors at universities with graduate programs. Without the REU, I would not have had the funding to be able to go to the Joint Mathematics Meetings or have anything to present at the Meetings.

I feel as though I am at a great advantage compared to the students at my institution who did not have an REU experience. During our senior thesis class, I had the advantage of already having completed research. I had experience with the process of picking a research question, learning the background, and completing meaningful research. During the REU we spent many hours learning how to present our progress. This made my presentations during my senior thesis much easier to prepare for and made me much more confidant presenting in front of my classmates. I felt ahead of my classmates who were unable to participate in an REU.

Also, the REU was an amazing experience outside of mathematics. The other students and I spent almost every night together playing card games, watching movies, or going to a local restaurant for trivia night. We even went to Cincinnati, Ohio one weekend. We got to visit the Newport Aquarium and explore the city. The nine of us became a close-knit group by the end of our eight weeks. It is nice to have students at other institutions to compare experiences. We all still keep in contact and visit each other.

**Matthew:**

Working in a team with other undergraduate researchers and in a program with other research teams provided new avenues for growth as well. I learned about doing mathematics in teams, something that I had little experience in, and how to efficiently divide labor while keeping everyone on the same page. By working with other students, I also gained appreciation for different student’s mathematical backgrounds and strengths. Talking with the other research teams also helped; their research exposed me to areas of combinatorics outside of what I was working in. Also, friendly competition between the groups helped motivate my team to keep going when we were stuck on a problem.

Through the REU, I gained more exposure to the larger mathematical community. Our program had regular speakers from other areas of the department and other schools come and talk, further exposing us to topics of current research. I learned about how the work I was doing fit in with research being done by other mathematicians (many with much more experience than me). The REU also allowed me to participate in the 2017 Joint Mathematics Meetings in Atlanta, Georgia in January where I was able to see the full breadth of current mathematics research beyond combinatorics. The other participants and I were able to learn more about graduate school by talking to our mentors and other mathematicians and attending the Graduate School Fair at the meetings in Atlanta.

**Stacie:**

When I declared my mathematics major my original goal was to eventually become an actuary. The summer after my freshmen year I did an internship at an actuary firm and realized that is not what I wanted to do for the rest of my life. Then, I decided I wanted to go to graduate school for pure mathematics, but I was unsure. After my REU experience I am confident that I have chosen the correct path. Overall, the Marshall University REU was unforgettable. I suggest that undergraduates apply to as many REUs as possible, because I think every student should have the opportunity to have this experience at some point before they graduate.

**Matthew:**

After participating in the REU, I feel as though I know so much more about mathematics. But even more, I feel as though I know more about how much there is out there that I don’t know. The program has motivated me to pursue a PhD in mathematics and do more mathematics research in the future, whether in combinatorics or another field. I suspect the frustration coming my way will be worth it.

]]>Today we celebrate the story of Marizza Bailey, who was honored last year by the White House with one of its Presidential Awards for Excellence in Mathematics and Science Teaching (PAEMST).

When Marizza Bailey was 12, her grandmother, Luz Mendizabal, came to live with her in California. Born in Peru, Luz put herself through graduate school to earn a doctorate degree in Mathematics Education, all while teaching full-time and raising eight children. She brought many things with her when she arrived in California: her voracious appetite for learning, her vast knowledge of mathematics, history and literature, but what Marizza appreciated most were the questions. “Why do you think that?” “What makes you say that?” “How do you know that?” A conversation with Luz was a series of questions and answers that stimulated critical thinking.

With a grandmother who was a mathematics teacher and who inspired thoughtful dialogue with her children and grandchildren, it’s no wonder that Marizza followed Luz’s footsteps and became a mathematics educator, as did Marizza’s mother before her. For this family, mathematics was not so much a career or a school subject, but a way of viewing and interpreting the world. Marizza says of her family, “They taught me by letting me wonder and allowing me to draw my own conclusions.”

When Marizza entered the classroom as an educator, she recognized the need for making mathematics relevant and relatable to students who didn’t share her specialized background. To do this, Marizza used materials from the American Mathematical Society (AMS) to supplement her curriculum. For students in her high school courses, this started with Mathematical Moments (posters that demonstrate an appreciation and understanding of the role mathematics plays in modern society). By printing out the classroom posters and having students listen to the accompanying podcasts, Marizza generated excitement around mathematical ideas that touched on real-life concepts. For older students who are considering studying mathematics in college, she enticed them with some of the more accessible articles in the Bulletin of the AMS. Last year, she took a group of students to the Joint Mathematics Meeting in Seattle, where they were able to interact with a host of professional mathematicians.

**What Went Right**

As an educator at BASIS Scottsdale, a charter school in Arizona, Marizza encountered a rare and enviable problem: what do you do with students who finish calculus and are hungry for more mathematics? This led her to create the course *Introduction to Categories*, the first of many unique post-calculus classes that Marizza would develop and teach to help meet her students’ desire for more advanced-level mathematics courses.

Each year, Marizza keeps a teaching journal where she records “what went right” and “what went wrong.” When she began teaching *Introduction to Categories*, typically a college-level course, she taught it in a typically college-level way: through lecture. This method quickly found itself in the “what went wrong” section of her journal. She opened the classroom up to more active learning, peer interaction, and discovery-centered lessons – the same strategies that had endeared her to mathematics as a child, learning from her grandmother – to vastly improved results. The success of *Introduction to Categories* allowed Marizza to develop more post-calculus courses: *Vector Calculus*, *Linear Algebra*, *History of Mathematics*, and *Complex Analysis*.

**Exceptional STEM Teachers Deserve to Be Recognized**

Each year, the National Science Foundation (NSF) has the honor of recognizing up to 108 teachers for the Presidential Awards for Excellence in Mathematics and Science Teaching (PAEMST). In 2015, Marizza Bailey was one of those educators.

NSF invited Marizza to Washington, D.C., for the National Recognition Event, which included two days of professional development opportunities, an awards ceremony featuring national leaders in STEM education policy and a visit to the White House. She also received a certificate signed by the President of the United States (U.S.), and a $10,000 award from NSF.

Marizza and the majority of Presidential Awardees say that this wasn’t the main perk of being an awardee. Rather, it was the connections they were able to make with educators and thought-leaders from across the country. At the recognition event, Marizza was able to network with peers from all 50 states and leading professional development facilitators. She also had the chance to meet a personal hero: Vi Hart, a mathematical YouTuber, who inspires many of her own students. Forging lasting relationships is inevitable when so many mathematics enthusiasts are brought together for a week of celebrating each other and the profession they love.

NSF and the PAEMST team believe that exceptional STEM teachers like Marizza deserve to be recognized. PAEMST are the nation’s highest honors for K-12 STEM teaching, and more than 4,700 teachers have received the award since the program began in 1983. Nominations and applications for K-6th grade teachers open this fall at www.paemst.org.

*The National Science Foundation administers the Presidential Awards for Excellence in Mathematics and Science Teaching (PAEMST) on behalf of The White House Office of Science and Technology Policy.

]]>When I first started incorporating active learning in the classroom, I struggled with getting my students to buy into being active. I made worksheets, put the students in groups, and excitedly set them off to discover and play with mathematical ideas. Despite this, many students were inclined to sit silently in a group of four and work on the problems on their own.

But really, who can blame them? First, this propensity towards solitude can be explained by basic human nature: specifically, the fear of being wrong. We don’t want to be wrong. At least, we don’t want to be wrong in front of other people. From that perspective, working alone is safe and comfortable. We should view our job as teachers as one of helping our students overcome this basic human inclination, as opposed to viewing it as a failure or shortcoming on their part.

Beyond this, the desire to work alone can be attributed to culture and expectations. Many students’ formative educational years have been spent sitting silently in desks passively absorbing lectures. If they feel this is what is expected of a math class, then it is natural for them to continue to sit silently, even if the environment is meant to be collaborative. Of course, it is not my intention to imply that this is an issue that is entirely the students’ fault – maybe my questions weren’t sufficiently open-ended, maybe I wasn’t doing a good enough job at “selling it,” maybe the students just like working alone, maybe, maybe, maybe… The list goes on.

I tried some of my standard tricks to foster communication among the students. I would prepare impassioned pep talks about the benefits of working with your peers. This technique flopped for obvious reasons – no one wants to listen to what they are told is good for them. Otherwise, cigarette companies and fast food restaurants would go bankrupt and I would be much more diligent about flossing. I’d try to lighten the mood, saying “this isn’t a library, you’re welcome to talk to one another.” I’d give a difficult problem and leave the room to get a drink of water, forcing the students to rely on one another. These strategies helped, but never served to create the classroom of my dreams – one where students discuss math problems at such a frenzied pace that time ceases to exist; one that causes passersby to wonder whether we are having a math class or developing some bizarre scientific improv comedy troupe.

Over time, I continued to reflect on my own teaching and sought advice from more experienced practitioners of active learning. As a result, I have developed a few strategies that have been effective in my classrooms. One of the most effective strategies for me has come from eliminating those pesky desks that keep getting in the way of my students’ learning.

One day after watching *Back to the Future* for the umpteenth time, Doc Brown gave me a spark of inspiration: “Roads? Where we’re going we don’t need roads!” But now replace “roads” with “desks.” Maybe the desks were the problem.

In the spring of 2014 I was teaching a Graph Theory class in my favorite classroom on campus. It has chalkboards on three of the walls, and the fourth wall is a bank of windows that looks out onto our campus quad, which is very pretty with a fountain and trees. It’s sort of a mathematician’s paradise. The most important aspect of this room is the board space – there’s enough room for 24 people to work on the board at the same time. I decided to test this desk-free learning idea: instead of sitting in their desks, what if my students spent most of their class time at the board? This had a noticeable impact on the quality of conversations and engagement in the classroom. I’ve found success in implementing this strategy in different ways in different classes. It has led to more dynamic exam review sessions for lower-level calculus and linear algebra classes, deeper learning in my introduction to proofs classes, and hotbeds of mathematical ideas in my graph theory classes. I’d like to offer some tips and tricks on how this can be done in a variety of settings.

I typically prepare a worksheet with problems for each class that are meant to guide the students through a certain topic or idea. If necessary, I start class with a short lecture introducing some new ideas or definitions and then set the students to work on the problems I have prepared, telling them that I want them to get up and work at the board in groups with their peers.

From a practical teaching perspective, there are a number of benefits to doing this. Because I can see what everyone is doing from just about anywhere (as long as the room is convex), I can easily assess each group’s progress from a vantage point in the center of the room and quickly determine which group needs the most immediate attention. If everyone seems to be making a common mistake, I can pull the group back together and add clarification or facilitate a large group discussion. Similarly, if one of the groups has something interesting to share – for example, an interesting example/counterexample or a solution that is different from what everyone else has done – it is easy to bring the whole class to their work area and let them share their ideas.

This practice can be particularly effective in a graph theory class where dynamic problem solving is so important. For example, solving problems about planar graphs on paper can be frustrating if you have to keep redrawing the same graph until you find a nice planar drawing. On the board, it is much easier to just erase the problematic edges without having start from scratch. In a different activity, I give the students pseudocode for an algorithm (for example, Dijkstra’s Algorithm), without telling them what the algorithm does. They decipher the pseudocode and run through the algorithm on an example graph to try to figure out what the algorithm does. After that, we discuss what the algorithm does and prove that it works as a group. This is easier once the students already have a solid intuitive understanding of what the algorithm does because it separates the difficulty of such proofs into more manageable pieces: first, understand what the algorithm does, and second understand techniques for proving an algorithm works.

Most importantly, being at the board helps students overcome the fear of being wrong. Because work on the board is inherently temporary, students don’t have the same reservations about writing down some ideas and sharing their thoughts, even if they are incomplete and especially if they may be wrong. I call this the “Bob Ross approach to mathematics.” On *The Joy of Painting*, Bob Ross taught us “We don’t make mistakes. We just make happy little accidents.” Overcoming their initial fear of making mistakes helps students get to the point where real learning happens.

Unlike working on a piece of paper, which carries a natural expectation that you will start writing at the top and finish writing at the bottom, work at the chalkboard can flow more organically. Students come to see that solutions often take serpentine paths through different parts of the problem and various examples until you figure out how to put all the pieces together. Then, once they find a solution, they can write out a clean version of the solution in their notebooks. In many cases, they just take pictures on their phones and transcribe their notes later.

Having students stand at the board also makes for a more social environment that naturally fosters collaboration and does seem to create a more active classroom. Standing helps everyone be more engaged, more physically active and, as a result, more mentally active. In a course evaluation, one student commented “Using the chalkboards in class was a great way to get our blood flowing and keep focused during class.” When you are standing, it is more reasonable to expect that you *should* be talking with the people around you.

As a result of spending more time collaborating with their peers, students come to see they are not alone in their confusion or struggles. They learn to ask questions, which can be as simple as “I didn’t get what you just said, can you say it again?” When I was in grad school I started asking that question, perhaps to the extent that people got tired of hearing it. It has been hugely beneficial to me and my students. When working together, students see different approaches and different ways of thinking about problems. As a result of having to answer questions posed by their peers, they reflect more deeply on the way they approach problems.

Heuristically, this also seems to help the students develop better problem solving skills. They realize that in order to solve a problem, it helps to write something, *anything*, to get your brain wrapped around the problem. Students learn to explore small examples, reflecting on their observations, and thinking about how to generalize those observations. By the end of the quarter, students who had initial reservations saying “do we have to work at the board?” have changed their attitude, with some saying “do we get to work at the board today?”

The tips I’ve discussed here are applicable beyond a course in Graph Theory and can be used beyond classes at a small liberal arts school with small classes. Having students work together to solve practice problems at the board without their notes or books can be valuable in helping them prepare for an exam. Teaching Assistants can implement similar practices in recitation sections with smaller groups of students, even if they are part of a large lecture course. At Seattle University, we don’t have a graduate program, but we have undergraduate Learning Assistants who facilitate study groups for our lower-level math courses. We train our LAs to facilitate group work in this way so that students are actively engaged during their study groups.

In mathematics, we often pride ourselves on the fact that our research can be done wherever there’s a chalkboard. We should strive to include this in the way we help our students learn!

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